0,5

From the 5th term on, all terms have a primitive divisor; in other words, a prime divisor that divides no earlier term in the sequence. A proof appears in the Everest-McLaren-Ward paper. - Graham Everest (g.everest(AT)uea.ac.uk), Oct 26 2005

Twelve prime terms are known, occurring at indices 4,5,6,7,8,11,13,16,43,52,206,647. The last two have been checked for probable primality only. The 647-th term has 18498 decimal digits. Possibly these are the only prime terms in the entire sequence. - Graham Everest (g.everest(AT)uea.ac.uk), Nov 28 2006

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. H. Buchholz and R. L. Rathbun, "An infinite set of Heron triangles with two rational medians", Amer. Math. Monthly, 104 (1997), 107-115.

G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; pp. 9, 179.

G. Everest et al., Primes generated by recurrence sequences, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431.

David Gale, Mathematical Entertainments: "The strange and surprising saga of the Somos sequence", Math. Intelligencer, 13(1) (1991), pp. 40-42.

A. N. W. Hone, Elliptic curves and quadratic recurrence sequences, Bulletin of the London Mathematical Society 37 (2005) 161-171.

J. L. Malouf, "An integer sequence from a rational recursion", Discr. Math. 110 (1992), 257-261.

R. M. Robinson, "Periodicity of Somos sequences", Proc. Amer. Math. Soc., 116 (1992), 613-619.

Alfred J. van der Poorten, Elliptic Curves and Continued Fractions, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.5.

Robert G. Wilson v, Table of a(n) for n = 0..100.

H. W. Braden, V. Z. Enolskii and A. N. W. Hone, Bilinear recurrences and addition formulae for hyperelliptic sigma functions

Graham Everest, Gerard Mclaren and Tom Ward, Primitive divisors of elliptic divisibility sequences 2005

G. Everest, S. Stevens, D. Tamsett and T. Ward, Primitive divisors of quadratic polynomial sequences

S. Fomin and A. Zelevinsky, The Laurent phenomemon

A. N. W. Hone, Sigma function solution of the initial value problem for Somos 5 sequences.

A. N. W. Hone, Algebraic curves, integer sequences and a discrete Painleve transcendent, Proceedings of SIDE 6, Helsinki, Finland, 2004.

J. Propp, The Somos Sequence Site

J. Propp, The 2002 REACH tee-shirt

M. Somos, Somos 6 Sequence

M. Somos, Brief history of the Somos sequence problem

D. E. Speyer, Perfect matchings and the octahedral recurrence

A. J. van der Poorten, Recurrence relations for elliptic sequences...

A. J. van der Poorten, Hyperelliptic curves, continued fractions and Somos sequences

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Index entries for two-way infinite sequences

a(n+1)/a(n) seems to be asymptotic to C^n with C=1.226....... - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 07 2002. Confirmed by Hone - see below.

The terms of the sequence have the leading order asymptotics log a(n) ~ D n^2 with D = zeta(w1)*k^2/(2*w1)-log|sigma(k)| = 0.10222281... where zeta and sigma are the Weierstrass functions with invariants g2 = 4, g3 = -1, w1 = 1.496729323 is the real half-period of the corresponding elliptic curve, k = -1.134273216 as above. This agrees with Benoit Cloitre's numerical result with C = exp(2D) = 1.2268447... - Andrew Hone (anwh(AT)kent.ac.uk), Feb 09 2005

a(n) = (a(n-1)*a(n-3) + a(n-2)^2)/a(n-4); a(0) = a(1) = a(2) = a(3) = 1; exact formula is a(n) = A*B^n*sigma (z_0+nk)/(sigma (k))^(n^2), where sigma is the Weierstrass sigma function associated to the elliptic curve y^2 = 4*x^3-4*x+1, A = 1/sigma(z_0) = 0.112724016-0.824911687*i, B = sigma(k)*sigma (z_0)/sigma (z_0+k) = 0.215971963+0.616028193*i, k = 1.859185431, z_0 = 0.204680500+1.225694691*i, sigma(k) = 1.555836426, all to 9 decimal places. This is a special case of a general formula for 4th order bilinear recurrences. The Somos-4 sequence corresponds to the sequence of points (2n-3)P on the curve, where P = (0, 1). - Andrew Hone (anwh(AT)kent.ac.uk), Oct 12 2005

Digits:=11; f(x):=4*x^3-4*x+1; sols:=evalf(solve(f(x), x)); e1:=Re(sols[1]); e3:=Re(sols[2]); w1:=evalf(Int((f(x))^(-0.5), x=e1..infinity)); w3:=I*evalf(Int((-f(x))^(-0.5), x=-infinity..e3)); k:=2*w1-evalf(Int((f(x))^(-0.5), x=1..infinity)); z0:=w3+evalf(Int((f(x))^(-0.5), x=e3..-1)); A:=1/WeierstrassSigma(z0, 4.0, -1.0); B:=WeierstrassSigma(k, 4.0, -1.0)/WeierstrassSigma(z0+k, 4.0, -1.0)/A; for n from 0 to 10 do a[n]:=A*B^n*WeierstrassSigma(z0+n*k, 4.0, -1.0)/(WeierstrassSigma(k, 4.0, -1.0))^(n^2) od; (Andrew Hone (anwh(AT)kent.ac.uk), Oct 12 2005)

a[0] = a[1] = a[2] = a[3] = 1; a[n_] := a[n] = (a[n - 1] a[n - 3] + a[n - 2]^2)/a[n - 4]; Array[a, 23] (* Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 04 2007 *)

Cf. A006721, A006722, A006723, A048736.

Cf. A028945, A028935, A151502.

For primes see A129739, A129740, A129741.

a(n)=(-1)^n*A006769(2n-3).

Cf. A165896. [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Sep 29 2009]

Sequence in context: A037231 A082449 A129741 this_sequence A084710 A088173 A129739

Adjacent sequences: A006717 A006718 A006719 this_sequence A006721 A006722 A006723

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

0,3

T(n,k) is the number of compatible k-sets of cluster variables in Fomin and Zelevinsky's Cluster algebra of finite type B_n. Take a row of this triangle regarded as a polynomial in x and rewrite as a polynomial in y := x+1. The coefficients of the polynomial in y give a row of triangle A008459 (squares of binomial coefficients). For example x^2+6*x+6=y^2+4*y+1. - Paul Boddington (psb(AT)maths.warwick.ac.uk), Mar 07 2003

T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e. bilateral Schroeder paths), having k N=(0,1) steps. E.g. T(2,0)=1 because we have DD; T(2,1)=6 because we have NED, NDE, EDN, END, DEN and DNE; T(2,2)=6 because we have NNEE, NENE, NEEN, EENN, ENEN and ENNE. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 20 2004

Another version of [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, . . .] DELTA [0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, . . . ] = 1; 1, 0; 1, 2, 0; 1, 6, 6, 0; 1, 12, 30, 20, 0; . . ., where DELTA is the operator defined in A084938 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr) Apr 15 2005

Terms in row n are the coefficients of the Legendre polynomial P(n,2x+1) with increasing powers of x.

Contribution from Peter Bala (pbala(AT)toucansurf.com), Oct 28 2008: (Start)

Row n of this triangle is the f-vector of the simplicial complex dual to an associahedron of type B_n (a cyclohedron)[Fomin & Reading, p.60]. See A008459 for the corresponding h-vectors for associahedra of type B_n and A001263 and A033282 respectively for the h-vectors and f-vectors for associahedra of type A_n.

An alternative description of this triangle in terms of f-vectors is as follows. Let A_n be the root lattice generated as a monoid by {e_i - e_j: 0 <= i,j <= n+1}. Let P(A_n) be the polytope formed by the convex hull of this generating set. Then the rows of this array are the f-vectors of a unimodular triangulation of P(A_n) [Ardila et al.]. A008459 is the corresponding array of h-vectors for these type A_n polytopes. See A127674 (without the signs) for the array of f-vectors for type C_n polytopes and A108556 for the array of f-vectors associated with type D_n polytopes.

The S-transform on the ring of polynomials is the linear transformation of polynomials that is defined on the basis monomials x^k by S(x^k) = binomial(x,k) = x(x-1)...(x-k+1)/k!. Let P_n(x) denote the S-transform of the n-th row polynomial of this array. In the notation of [Hetyei] these are the Stirling polynomials of the type B associahedra. The first few values are P_1(x) = 2*x + 1, P_2(x) = 3*x^2 + 3*x + 1 and P_3(x) = (10*x^3 + 15*x^2 + 11*x + 3)/3. These polynomials have their zeros on the vertical line Re x = -1/2 in the complex plane, that is, the polynomials P_n(-x) satisfy a Riemann hypothesis. See A142995 for further details. The sequence of values P_n(k) for k = 0,1,2,3, ... produces the n-th row of A108625. (End)

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 366.

S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations, J. Amer. Math. Soc. 15 (2002) no. 2, 497-529

S. Fomin and A. Zelevinsky, Y-Systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 977-1018.

R. A. Sulanke, Objects counted by the central Delannoy numbers. J. Integer Seq. 6 (2003), no. 1, Article 03.1.5.

Gabor Hetyei. The Stirling polynomial of a simplicial complex. Discrete and Computational Geometry 35, Number 3, March 2006, pp 437-455. [From Peter Bala (pbala(AT)toucansurf.com), Oct 28 2008]

T. D. Noe, Rows n=0..100 of triangle, flattened

S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations, J. Amer. Math. Soc. 15 (2002), no. 2, 497-529.

S. Fomin and A. Zelevinsky, Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 977-1018.

R. A. Sulanke, Objects counted by the central Delannoy numbers., J. Integer Seq. 6 (2003), no. 1, Article 03.1.5.

V. Strehl, Recurrences and Legendre transform

F. Chapoton, Enumerative properties of generalized associahedra

F. Ardila, M. Beck, S. Hosten, J. Pfeifle and K. Seashore, Root polytopes and growth series of root lattices [From Peter Bala (pbala(AT)toucansurf.com), Oct 28 2008]

S. Fomin, N. Reading, Root systems and generalized associahedra, Lecture notes for IAS/Park-City 2004. [From Peter Bala (pbala(AT)toucansurf.com), Oct 28 2008]

G. Hetyei, Face enumeration using generalized binomial coefficients. This is the draft version of Hetyei's paper referenced above. [From Peter Bala (pbala(AT)toucansurf.com), Oct 28 2008]

T(n, k) = (n+k)!/(k!^2*(n-k)!) = T(n-1, k)*(n+k)/(n-k) = T(n, k-1)*(n+k)*(n-k+1)/k^2 = T(n-1, k-1)*(n+k)*(n+k-1)/k^2.

G.f.=G(t, z)=1/sqrt(1-2z-4tz+z^2). Row generating polynomials=P_n(1+2z), i.e. T(n, k)=[z^k]P_n(1+2z), where P_n are the Legendre polynomials. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 20 2004

Sum_{k>=0} T(n, k)*A000172(k) = Sum_{k>=0} T(n, k)^2 = A005259(n). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 08 2005 - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 08 2005

1; 1,2; 1,6,6; 1,12,30,20; 1,20,90,140,70; ...

with(orthopoly): seq([1, seq(coeff(expand(P(n, 1+2*z)), z^k), k=1..n)], n=0..9);

(PARI) T(n, k)=local(t); if(n<0, 0, t=(x+x^2)^n; for(k=1, n, t=t'); polcoeff(t, k)/n!)

Columns include A000012, A002378, A033487 on the left and A000984, A002457, A002544 on the right. Main diagonal is A006480. Row sums are A001850.

Cf. A008459.

Cf. A104684

A033282 (f-vectors type A associahedra), A108625, A080721 (f-vectors type D associahedra) [From Peter Bala (pbala(AT)toucansurf.com), Oct 28 2008]

Sequence in context: A133314 A049019 A046651 this_sequence A089231 A052296 A019538

Adjacent sequences: A063004 A063005 A063006 this_sequence A063008 A063009 A063010

nonn,tabl,nice,easy

Henry Bottomley (se16(AT)btinternet.com), Jul 02 2001

More terms from Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 08 2005

0,7

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. H. Buchholz and R. L. Rathbun, "An infinite set of Heron triangles with two rational medians", Amer. Math. Monthly, 104 (1997), 107-115.

David Gale, "The strange and surprising saga of the Somos sequence", Math. Intelligencer 13(1) (1991), pp. 40-42.

J. L. Malouf, "An integer sequence from a rational recursion", Discr. Math. 110 (1992), 257-261.

C. Pickover, Mazes for the Mind, St. Martin's Press, NY, 1992, p. 350.

R. M. Robinson, "Periodicity of Somos sequences", Proc. Amer. Math. Soc., 116 (1992), 613-619.

T. D. Noe, Table of n, a(n) for n=0..100

Index entries for two-way infinite sequences

S. Fomin and A. Zelevinsky, The Laurent phenomemon

M. Somos, Somos 6 Sequence

M. Somos, Brief history of the Somos sequence problem

A. van der Poorten, Hyperelliptic curves, continued fractions and Somos sequences

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Cf. A006720, A006721, A006723, A048736.

Sequence in context: A146275 A089636 A083366 this_sequence A039774 A114001 A004044

Adjacent sequences: A006719 A006720 A006721 this_sequence A006723 A006724 A006725

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Aug 22 2000

0,2

Also Schoenheim bound L_1(n,5,4).

Degrees of polynomials defined by p(n)=(x^(n+1)*p(n-1)p(n-3)+p(n-2)^2)/p(n-4), p(-4)=p(-3)=p(-2)=p(-1)=1. - Michael Somos, Jul 21 2004

Degrees of polynomial tau-functions of q-discrete Painleve I, which generate sequence A095708 when q=2 (up to an offset of 3). - Andrew Hone (anwh(AT)kent.ac.uk), Jul 29 2004

Because of the Laurent phenomenon for the general q-discrete Painleve I tau-function recurrence p(n)=(a*x^(n+1)*p(n-1)*p(n-3)+b*p(n-2)^2)/p(n-4), p(n) for n>-1 will always be a polynomial in x and a Laurent polynomial in a,b and the initial data p[ -4],p[ -3],p[ -2],p[ -1]. - Andrew Hone (anwh(AT)kent.ac.uk), Jul 29 2004

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

H. R. Henze and C. M. Blair, The number of structurally isomeric hydrocarbons of the ethylene series, J. Amer. Chem. Soc., 55 (1933), 680-685.

W. H. Mills and R. C. Mullin, Coverings and packings, pp. 371-399 of J. H. Dinitz and D. R. Stinson, editors,a Contemporary Design Theory, Wiley, 1992. See Eq. 1.

L. Smiley, Hidden Hexagons, (preprint)

S. Fomin and A. Zelevinsky, The Laurent phenomenon, Advances in Applied Mathematics 28 (2002) 119-144.

Brian OSullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 10a, lambda=3]

Index entries for two-way infinite sequences

Index entries for covering numbers

A. N. W. Hone, Algebraic curves, integer sequences and a discrete Painleve transcendent, Proceedings of SIDE 6, Helsinki, Finland, 2004. [Set a(n)=d(n+3) on p. 8]

G.f.: 1/((1-x)^3*(1-x^3)). a(n)=-a(-6-n)=3a(n-1)-3a(n-2)+2a(n-3)-3a(n-4)+3a(n-5)-a(n-6).

The simplest recurrence is fourth order: a(n)=a(n-1)+a(n-3)-a(n-4)+n+1, which gives the G.f. 1/((1-x)^3(1-x^3)), with a(-4)=a(-3)=a(-2)=a(-1)=0. An explicit formula is a(n)=n^3/18+n^2/2+4*n/3+1+2/(9*sqrt(3))*sin(2*Pi*n/3). - Andrew Hone (anwh(AT)kent.ac.uk), Jul 29 2004

a(n)=[2*A000027(n+1)+3*A000292(n+1)+A049347(n-1)+1+3*A000217(n+1)]/9. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 16 2007

Polynomials p(0)=x+1, p(1)=x^3+x^2+1, p(2)=x^6+x^5+x^3+x^2+2x+1, ...

L := proc(v, k, t, l) local i, t1; t1 := l; for i from v-t+1 to v do t1 := ceil(t1*i/(i-(v-k))); od: t1; end; # gives Schoenheim bound L_l(v, k, t). Current sequence is L_1(n, n-3, n-4, 1).

(PARI) a(n)=if(n<-5, -a(-6-n), polcoeff(1/(1-x)^3/(1-x^3)+x^n*O(x), n)) /* Michael Somos, Jul 21 2004 */

Cf. A014126, A000631.

A column of A036838.

Cf. A095708.

Sequence in context: A140126 A140235 A010000 this_sequence A147456 A011849 A095944

Adjacent sequences: A014122 A014123 A014124 this_sequence A014126 A014127 A014128

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com).

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Dec 24 1999

0,6

Using the addition formula for the Weierstrass sigma function it is simple to prove that the subsequence of even terms of a Somos-5 type sequence satisfy a 4th order recurrence of Somos-4 type and similarly the odd subsequence satsifies the same 4th order recurrence. - Andrew Hone (anwh(AT)kent.ac.uk), Aug 24 2004

log(a(n)) ~ 0.071626946 * n^2. (Hone)

R. H. Buchholz and R. L. Rathbun, "An infinite set of Heron triangles with two rational medians", Amer. Math. Monthly, 104 (1997), 107-115.

David Gale, "The strange and surprising saga of the Somos sequence", Math. Intelligencer 13(1) (1991), pp. 40-42.

A. N. W. Hone, Elliptic curves and quadratic recurrence sequences, Bull. Lond. Math. Soc. 37 (2005) 161-171.

J. L. Malouf, "An integer sequence from a rational recursion", Discr. Math. 110 (1992), 257-261.

R. M. Robinson, "Periodicity of Somos sequences", Proc. Amer. Math. Soc., 116 (1992), 613-619.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alfred J. van der Poorten, Elliptic Curves and Continued Fractions, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.5.

T. D. Noe, Table of n, a(n) for n=0..100

Index entries for two-way infinite sequences

S. Fomin and A. Zelevinsky, The Laurent phenomemon

A. N. W. Hone, Sigma function solution of the initial value problem for Somos 5 sequences

J. Propp, The Somos Sequence Site

J. Propp, The 2002 REACH tee-shirt

M. Somos, Somos 6 Sequence

M. Somos, Brief history of the Somos sequence problem

D. E. Speyer, Perfect matchings and the octahedral recurrence

A. J. van der Poorten, Elliptic curves and continued fractions

A. J. van der Poorten, Recurrence relations for elliptic sequences...

A. J. van der Poorten, Hyperelliptic curves, continued fractions and Somos sequences

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

D. Zagier, Problems posed at the St Andrews Colloquium, 1996

Comments from Andrew Hone (anwh(AT)kent.ac.uk), Aug 24 2004: "Both the even terms b(n)=a(2n) and odd terms b(n)=a(2n+1) satisfy the fourth order recurrence b(n)=(b(n-1)*b(n-3)+8*b(n-2)^2)/b(n-4).

"Hence the general formula is a(2n)=A*B^n*sigma(c+n*k)/sigma(k)^(n^2), a(2n+1)=D*E^n*sigma(f+n*k)/sigma(k)^(n^2) where sigma is the Weierstrass sigma function associated to the elliptic curve y^2=4*x^3-(121/12)*x+845/216 (this is birationally equivalent to the minimal model V^2+U*V+6*V=U^3+7*U^2+12*U given by van der Poorten).

"The real/imaginary half-periods of the curve are w1=1.181965956, w3=0.973928783*I and the constants are A=0.142427718-1.037985022*I, B=0.341936209+0.389300717*I, c=0.163392411+w3, k=1.018573545, D=-0.363554228-0.803200610*I, E=0.644801269+0.734118205*I, f=c+k/2-w1 all to 9 decimal places."

Cf. A006720, A006722, A006723, A048736.

Cf. A006720.

Sequence in context: A124561 A167604 A065510 this_sequence A111289 A127181 A113734

Adjacent sequences: A006718 A006719 A006720 this_sequence A006722 A006723 A006724

easy,nonn,nice

N. J. A. Sloane (njas(AT)research.att.com).

3,3

T(n+3,k) is also the number of compatible k-sets of cluster variables in Fomin and Zelevinsky's cluster algebra of finite type A_n. Take a row of this triangle regarded as a polynomial in x and rewrite as a polynomial in y := x+1. The coefficients of the polynomial in y give a row of the triangle of Narayana numbers A001263. For example x^2+5*x+5=y^2+3*y+1. - Paul Boddington (psb(AT)maths.warwick.ac.uk), Mar 07 2003

Number of standard Young tableaux of shape (k+1,k+1,1^(n-k-3)), where 1^(n-k-3) denotes a sequence of n-k-3 1's (see the Stanley reference).

Number of k dimensional 'faces' of the n dimensional associahedron (see Simion, p. 168). - Mitch Harris (maharri(AT)gmail.com), Jan 16 2007

Mirror image of triangle A126216 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 19 2007

For relation to Lagrange inversion or series reversion and the geometry of associahedra or Stasheff polytopes (and other combinatorial objects) see A133437. [From Tom Copeland (tcjpn(AT)msn.com), Sep 29 2008]

Row generating polynomials 1/(n+1)*Jacobi_P(n,1,1,2*x+1). Row n of this triangle is the f-vector of the simplicial complex dual to an associahedron of type A_n [Fomin & Reading, p.60]. See A001263 for the corresponding array of h-vectors for associahedra of type A_n. See A063007 and A080721 for the f-vectors for associahedra of type B and type D respectively. [From Peter Bala (pbala(AT)toucansurf.com), Oct 28 2008]

D. Beckwith, Legendre polynomials and polygon dissections?, Amer. Math. Monthly, 105 (1998), 256-257.

A. Cayley, On the partitions of a polygon, Proc. London Math. Soc., 22 (1891), 237-262 = Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 93ff. (See p. 239.)

B. Drake, I. M. Gessel and G. Xin, Three proofs and a generalization of the Goulden-Litsyn-Shevelev conjecture ..., J. Integer Sequences, Vol. 10 (2007), #07.3.7.

P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 1999, 203-229.

S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations, J. Amer. Math. Soc. 15 (2002) no.2, 497-529.

S. Fomin and A. Zelevinsky, Y-Systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 977-1018.

G. Kreweras, Sur les partitions..., Discrete Math. 1 (1972), 333-350.

R. C. Read, On general dissections of a polygon, Aequat. Math. 18 (1978), 370-388.

R. Simion, "Convex Polytopes and Enumeration", Adv. in Appl. Math. 18 (1997) pp. 149-180.

R. P. Stanley, Polygon dissections and standard Young tableaux, J. Comb. Theory, Ser. A, 76, 175-177, 1996.

Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4. [From Tom Copeland (tcjpn(AT)msn.com), Nov 03 2008]

F. Chapoton, Enumerative properties of generalized associahedra

P. Flajolet and M. Noy, Analytic Combinatorics of Non-crossing Configurations, Discrete Math., 204, 1999, 203-229.

S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations, J. Amer. Math. Soc. 15 (2002), no. 2, 497-529.

S. Fomin and A. Zelevinsky, Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 977-1018.

R. C. Read, On general dissections of a polygon, Aequat. Math. 18 (1978), 370-388.

R. P. Stanley, Polygon dissections and standard Young tableaux, J. Comb. Theory, Ser. A, 76, 175-177, 1996.

S. Fomin, N. Reading, Root systems and generalized associahedra, Lecture notes for IAS/Park-City 2004. [From Peter Bala (pbala(AT)toucansurf.com), Oct 28 2008]

G.f. G=G(t, z) satisfies (1+t)G^2-z(1-z-2tz)G+tz^4=0.

T(n, k)=binomial(n-3, k)*binomial(n+k-1, k)/(k+1) for n >= 3, 0 <=k <=n-3.

Contribution from Tom Copeland (tcjpn(AT)msn.com), Nov 03 2008: (Start)

Two g.f.s (f1 and f2) for A033282 and their inverses (x1 and x2) can be derived from the Drake and Barry references.

1. a: f1(x,t) = y = {1 - (2t+1) x - sqrt[1 - (2t+1) 2x + x^2]}/[2x (t+1)]

= t x + (t + 2 t^2) x^2 + (t + 5 t^2 + 5 t^3) x^3 + ...

b: x1 = y/[t + (2t+1)y + (t+1)y^2] = y {1/[t/(t+1) + y] - 1/(1+y)}

= (y/t) - (1+2t)(y/t)^2 + (1+ 3t + 3t^2)(y/t)^3 +...

2. a: f2(x,t) = y = {1 - x - sqrt[(1-x)^2 - 4xt]}/[2(t+1)]

= (t/(t+1)) x + t x^2 + (t + 2 t^2) x^3 + (t + 5 t^2 + 5 t^3) x^4 + ...

b: x2 = y(t+1) [1- y(t+1)]/[t + y(t+1)]

= (t+1) (y/t) - (t+1)^3 (y/t)^2 + (t+1)^4 (y/t)^3 + ...

c: y/x2(y,t) = [t/(t+1) + y] / [1- y(t+1)]

= t/(t+1) + (1+t) y + (1+t)^2 y^2 + (1+t)^3 y^3 + ...

x2(y,t) can be used along with the Lagrange inversion for an o.g.f. (A133437)

to generate A033282 and show that A133437 is a refinement of A033282,

i.e., a refinement of the f-polynomials of the associahedra, the Stasheff polytopes.

y/x2(y,t) can be used along with the indirect Lagrange inversion (A134264)

to generate A033282 and show that A134264 is a refinement of A001263, i.e.,

a refinement of the h-polynomials of the associahedra.

f1[x,t](t+1) gives a generator for A088617.

f1[xt,1/t](t+1) gives a generator for A060693, with inverse y/[1 + t + (2+t) y + y^2].

f1[x(t-1),1/(t-1)]t gives a generator for A001263, with inverse y/[t + (1+t) y + y^2].

The unsigned coefficients of x1(y t,t) are A074909, reverse rows of A135278. (End)

G.f.: 1/(1-xy-(x+xy)/(1-xy/(1-(x+xy)/(1-xy/(1-(x+xy)/(1-xy/(1-.... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), Feb 06 2009]

1; 1,2; 1,5,5; 1,9,21,14; 1,14,56,84,42;

Diagonals : A000012, A000096, A033275, A033276, A033277, A033278, A033279; A000108, A002054, A002055, A002056, A007160, A033280, A033281 Row sums : A001003 (Schroeder numbers, first term omitted) . See A086810 for another version.

A007160 is a diagonal. Cf. A001263.

With leading zero: A086810.

Cf. A019538 'faces' of the permutohedron.

Cf. A063007 (f-vectors type B associahedra), A080721 (f-vectors type D associahedra), A126216 (mirror image). [From Peter Bala (pbala(AT)toucansurf.com), Oct 28 2008]

Sequence in context: A145882 A111785 A021468 this_sequence A126350 A079502 A126124

Adjacent sequences: A033279 A033280 A033281 this_sequence A033283 A033284 A033285

nonn,tabl,easy

N. J. A. Sloane (njas(AT)research.att.com).

Added a missing factor of 2 for expansions of f1 and f2 Tom Copeland (tcjpn(AT)msn.com), Apr 12 2009

1,2

Number of partitions with Ferrers plots that fit inside an n X n box, but not in an n-1 X n-1 box. - Wouter Meeussen (wouter.meeussen(AT)pandora.be), Dec 10 2001

From Benoit Cloitre, Jan 29 2002: Let m(1,j)=j, m(i,1)=i and m(i,j)=m(i- 1,j)+m(i,j-1); then a(n) = m(n,n):

1 2 3 4 .....

2 4 7 11 .......

3 7 14 25 .......

4 11 25 50 .......

This sequence also gives the number of clusters and non-crossing partitions of type D_n. - Frederic Chapoton (fchapoton(AT)voila.fr), Jan 31 2005

If Y is a 2-subset of a 2n-set X then a(n) is the number of (n+1)-subsets of X intersecting Y. - Milan R. Janjic (agnus(AT)blic.net), Nov 18 2007

Prefaced with a 1: (1, 1, 4, 14, 50,...) and convolved with the Catalan sequence = A097613: (1, 2, 7, 25, 91,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 15 2009]

Fomin, Sergey and Zelevinsky, Andrei, Y-systems and generalized associahedra, Ann. of Math. (2) 158,2003.

Hugh Thomas, math.CO/0311334: Tamari Lattices and Non-Crossing Partitions in Types B and D.

F. Chapoton, Clusters.

Milan Janjic, Two Enumerative Functions

a(n+1)=binomial(2*n, n)+2*sum(i=0, n-1, binomial(n+i, i) (V's in Pascal's Triangle) Jon Perry (perry(AT)globalnet.co.uk) Apr 13 2004

a(n) = n*C(n-1) - (n-1)*C(n-2), where C(n) = A000108(n) = Catalan(n). For example, a(5) = 50 = 5*C(4) - 4*C(3) - 5*14 - 3*5 = 70 - 20. Triangle A128064 as an infinite lower triangular matrix * A000108 = A051924 prefaced with a 1: (1, 1, 4, 14, 50, 182,...) Gary W. Adamson (qntmpkt(AT)yahoo.com), May 15 2009

Sum of 3 central terms of Pascal's triangle: 2*C(2+2*n, n)+C(2+2*n, 1+n). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 20 2005

a(n+1)=A051597(2n,n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 26 2006

Contribution from Paul Barry (pbarry(AT)wit.ie), Oct 17 2009: (Start)

The sequence 1,1,4,... has a(n)=C(2n,n)-C(2(n-1),n-1)=0^n+sum{k=0..n, C(n-1,k-1)*A002426(k)}, and g.f. given by

(1-x)/(1-2x-2x^2/(1-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-.... (continued fraction). (End)

Sums of {1}, {2, 1, 1}, {2, 2, 3, 3, 2, 1, 1}, {2, 2, 4, 5, 7, 6, 7, 5, 5, 3, 2, 1, 1}, ...

C:=proc(n) options operator, arrow: binomial(2*n, n)/(n+1) end proc: seq(n*C(n-1)-(n-1)*C(n-2), n=2..25); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 08 2008

Z:=(1-z-sqrt(1-4*z))/sqrt(1-4*z): Zser:=series(Z, z=0, 32): seq(coeff(Zser, z, n), n=1..24); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 01 2007

Left-central elements of the (1, 2)-Pascal triangle A029635.

Cf. A000108, A024482 (diagonal from 2), A076540 (diagonal from 3), A000124 (row from 2), A004006 (row from 3), A006522 (row from 4).

Cf. A128064.

Sequence in context: A047065 A055990 A087945 this_sequence A076024 A062807 A117421

Adjacent sequences: A051921 A051922 A051923 this_sequence A051925 A051926 A051927

easy,nice,nonn

Barry E. Williams, Dec 19 1999

Edited by N. J. A. Sloane (njas(AT)research.att.com), May 03 2008, at the suggestion of R. J. Mathar.

0,8

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. H. Buchholz and R. L. Rathbun, "An infinite set of Heron triangles with two rational medians", Amer. Math. Monthly, 104 (1997), 107-115.

David Gale, "The strange and surprising saga of the Somos sequence", Math. Intelligencer 13(1) (1991), pp. 40-42.

J. L. Malouf, "An integer sequence from a rational recursion", Discr. Math. 110 (1992), 257-261.

R. M. Robinson, "Periodicity of Somos sequences", Proc. Amer. Math. Soc., 116 (1992), 613-619.

T. D. Noe, Table of n, a(n) for n=0..100

Index entries for two-way infinite sequences

S. Fomin and A. Zelevinsky, The Laurent phenomemon

J. Propp, The Somos Sequence Site

M. Somos, Somos 6 Sequence

M. Somos, Brief history of the Somos sequence problem

A. van der Poorten, Hyperelliptic curves, continued fractions and Somos sequences

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

(PARI) a(n)=if(n<0, a(6-n), if(n<7, 1, (a(n-1)*a(n-6)+a(n-2)*a(n-5)+a(n-3)*a(n-4))/a(n-7)))

Cf. A006720, A006721, A006722, A048736.

Sequence in context: A099170 A018095 A003217 this_sequence A096390 A092264 A135729

Adjacent sequences: A006720 A006721 A006722 this_sequence A006724 A006725 A006726

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Aug 22 2000

0,3

A second order recurrence with the Laurent property. This property is satisfied by any second order recurrence of the form a(n+2)=f(a(n+1))/a(n) where f is a polynomial of the form f(x)=x^m*p(x) with m a positive integer and p arbitrary. So if p has integer coefficients and a(0)=a(1)=1 then a(n) is an integer for all n.

As n tends to infinity, log(log(a(n)))/n tends to log((3+sqrt(5))/2) or about 0.962.

The Laurent property is satisfied by any second order recurrence of the form a(n+2)=f(a(n+1))/a(n) where f is a polynomial of the form f(x)=x^m*p(x) with m a positive integer greater than or equal to 2 and p arbitrary. In that case a(0)=a(1)=1 generates a sequence of integers and the ratios a(n+1)/a(n) and a(n+1)*a(n-1)/a(n)^2 are integers for all n. - Andrew Hone (anwh(AT)kent.ac.uk), Dec 12 2005

S. Fomin and A. Zelevinsky, The Laurent phenomenon, Advances in Applied Math. 28 (2002), 119-144.

a[0]:=1; a[1]:=1; f(x):=x^3+x^2; for n from 0 to 8 do a[n+2]:=simplify(subs(x=a[n+1], f(x))/a[n]) od; s[3]:=ln(2^2*3); s[4]:=ln(2^3*3^2*13); for n from 3 to 10000 do s[n+2]:=evalf(3*s[n+1]+ln(1+exp(-s[n+1]))-s[n]): od: print(evalf(ln(s[10002])/(10002))): evalf(ln((3+sqrt(5))/2)); # s[n]=ln(a[n]); ln(s[n])/n converges slowly to 0.962...

Sequence in context: A061149 A129933 A064320 this_sequence A058975 A057120 A112512

Adjacent sequences: A112370 A112371 A112372 this_sequence A112374 A112375 A112376

nonn

Andrew Hone (anwh(AT)kent.ac.uk), Dec 02 2005

1,3

Any sequence a(1),a(2),a(3),... defined by the recurrence a(n) = (a(n-1)+1)/a(n-2) (for n>2) has period 5. The theory of cluster algebras currently being developed by Fomin and Zelevinsky gives a context for these facts, but it doesn't really explain them in an elementary way. - James Propp, Nov 20, 2002

Terms of the simple continued fraction of 34/[sqrt(2405)-29]. Decimal expansion of 1248/11111. [From Paolo P. Lava (ppl(AT)spl.at), Aug 05 2009]

Sergey Fomin and Andrei Zelevinsky, Cluster algebras II: Finite type classification

a(n)=1/50*{19*(n mod 5)+19*[(n+1) mod 5]-[(n+2) mod 5]-[(n+3) mod 5]+9*[(n+4) mod 5]} - Paolo P. Lava (ppl(AT)spl.at), Nov 27 2006

a := 1; b := 1; f := proc(n) option remember; global a, b; if n=1 then a elif n=2 then b else (f(n-1)+1)/f(n-2); fi; end;

Cf. A076840, A076841, A076844, A076823.

Sequence in context: A083279 A159455 A105734 this_sequence A092542 A026552 A086437

Adjacent sequences: A076836 A076837 A076838 this_sequence A076840 A076841 A076842

nonn

N. J. A. Sloane (njas(AT)research.att.com), Nov 21 2002