0,1

Number of ways of writing n as a product of primes.

Number of ways of writing n as a sum of distinct powers of 2.

Continued fraction for golden ratio A001622.

Partial sums of A000007 (characteristic function of 0). - Jeremy Gardiner (jeremy.gardiner(AT)btinternet.com), Sep 08 2002

An example of an infinite sequence of positive integers whose distinct pairwise concatenations are all primes! - Don Reble, Apr 17 2005

Binomial transform of A000007; inverse binomial transform of A000079 . Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jul 07 2005

A063524(a(n)) = 1. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 11 2008]

For n >= 0, let M(n) be the matrix with 1st row = (n n+1) and 2nd row = (n+1 n+2). Then a(n) = absolute value of det(M(n)). [From Kailasam Viswanathan Iyer (kvi(AT)nitt.edu), Apr 11 2009]

The partial sums give the natural numbers (A000027). [From Daniel Forgues (squid(AT)zensearch.com), May 08 2009]

Contribution from Barbarel Tres Mil (barbarel3000(AT)yahoo.es), Sep 04 2009: (Start)

a(n) is also tau_1(n) where tau_2(n) is A000005

a(n) is a completely multiplicative arithmetical function.

a(n) is both square free and a perfect square. See A005117 and A000290. (End)

Also smallest divisor of n. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Sep 07 2009].

a(n) is also the decimal expansion of 10/9 [From Barbarel Tres Mil (barbarel3000(AT)yahoo.es), Sep 18 2009]

a(n) is also the number of complete graphs on n nodes. [From Pablo Chavez (pchavez(AT)cmu.edu), Sep 15 2009]

Totally multiplicative sequence with a(p) = 1 for prime p. Totally multiplicative sequence with a(p) = a(p-1) for prime p. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Oct 18 2009]

nth prime minus phi(prime(n)); number of divisors of n-th prime minus number of perfect partitions of n-th prime; the number of perfect partitions of n-th prime number; the number of perfect partitions of n-th non-composite number. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Oct 26 2009]

Contribution from Harlan J. Brothers (harlan(AT)brotherstechnology.com), Nov 01 2009: (Start)

For all n>0, the sequence of limit values for a(n)=n!Sum[k=n..inf, k/(k+1)! ]

Also, for all n != 0, a(n)=n^0 (End)

a(n) is also the number of 0-regular graphs on n vertices. [From Jason Kimberley (Jason.Kimberley(AT)newcastle.edu.au), Nov 07 2009]

Differences between consecutive n. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Dec 05 2009]

Triangle of numerators in Leibniz harmonic triangle. [From Paul Muljadi (paulmuljadi(AT)yahoo.com), Jan 21 2010]

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

Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

N. J. A. Sloane, Table of n, a(n) for n = 0..1000 [Useful when plotting one sequence against another. See Swayne link.]

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Index entries for sequences related to linear recurrences with constant coefficients

Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors ..., J. Integer Seqs., Vol. 6, 2003.

N. J. A. Sloane, Illustration of initial terms

D. F. Swayne, Plot pairs of sequences in the OEIS

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

Eric Weisstein's World of Mathematics, Chromatic Number

Eric Weisstein's World of Mathematics, Graph Cycle

G. Xiao, Contfrac

Index entries for "core" sequences

Index entries for characteristic functions

Index entries for continued fractions for constants

Index entries for related partition-counting sequences

Harlan Brothers, Factorial: Summation (formula 06.01.23.0002), The Wolfram Functions Site [From Harlan J. Brothers (harlan(AT)brotherstechnology.com), Nov 01 2009]

Wikipedia, Leibniz harmonic triangle [From Paul Muljadi (paulmuljadi(AT)yahoo.com), Jan 21 2010]

G.f.: 1/(1-x); a(n)=1. E.g.f.: e^x.

G.f.: Product[(1+x^(2^k)),{k,0,Infinity}]. - Zak Seidov (zakseidov(AT)yahoo.com), Apr 06 2007

Multiplicative with a(p^e) = 1.

Dirichlet generating function: zeta(s). - Franklin T. Adams-Watters, Sep 11 2005.

Regarded as a square array by antidiagonals, g.f. 1/((1-x)(1-y)), e.g.f. sum T(n,m) x^n/n! y^m/m! = e^{x+y}, e.g.f. sum T(n,m) x^n y^m/m! = e^y/(1-x). Regarded as a triangular array, g.f. 1/((1-x)(1-xy)), e.g.f. sum T(n,m) x^n y^m/m! = e^{xy}/(1-x). - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Feb 06 2006

Contribution from Barbarel Tres Mil (barbarel3000(AT)yahoo.es), Sep 04 2009: (Start)

a(n)=Sum(d|n,mu(n/d)*tau_2(d))=1, where tau_2(n)=A000005 and mu(n)=A008683

a(n)=|Sum(d|n,mu(d)*tau_2(d))|=1 (End)

a(n)=A000027(n)-A001477(n). - Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Oct 09 2009

a(n)=A002033(A000040(n))=A002033(A008578(n))=A000005(A000040(n))-A002033(n)=A000027(A000040(n))-A000010(A000040(n)). [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Oct 26 2009]

a(n)=A000027(n+1)-A000027(n). [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Dec 05 2009]

1.618033988749894848204586834... = 1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + ...)))) [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 14 2009]

A000012 := n->1;

[ seq(1, i=0..100) ];

a[n_] := 1

Array[1 &, 50] - Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 26 2006

Table[n!Sum[k/(k+1)!, {k, n, \[Infinity]}], {n, 10}] [From Harlan J. Brothers (harlan(AT)brotherstechnology.com), Nov 01 2009]

(MAGMA) [ 1 : n in [0..100]];

(PARI) a(n)=1

(PARI) { default(realprecision, 1080); phi = (1 + sqrt(5))/2; x=contfrac(phi); for (n=1, 1001, write("b000012.txt", n-1, " ", x[n])); } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 14 2009]

Cf. A000004, A007395, A010701.

Cf. A000027.

Cf. tau_2(n): A000005, tau_3(n): A007425, tau_4(n): A007426, tau_5(n): A061200, tau_6(n): A034695, (unordered) 2-factorizations of n: A038548, (unordered) 3-factorizations of n: A034836, A001055, (tau<=)_2(n): A006218, (tau<=)_3(n): A061201, (tau<=)_5(n): A061203, (tau<=)_6(n): A061204. [From Barbarel Tres Mil (barbarel3000(AT)yahoo.es), Sep 04 2009]

Cf. A000010, A000040, A002033, A008578. [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Oct 26 2009]

Regular graphs A005176 (any degree), A051031 (triangular array), chosen degrees: A000012 (k=0), A059841 (k=1), A008483 (k=2), A005638 (k=3), A033301 (k=4), A165626 (k=5), A165627 (k=6), A165628 (k=7). [From Jason Kimberley (Jason.Kimberley(AT)newcastle.edu.au), Nov 07 2009]

Sequence in context: A087960 A164660 A114523 this_sequence A008836 A064179 A106400

Adjacent sequences: A000009 A000010 A000011 this_sequence A000013 A000014 A000015

core,easy,nonn,mult,cofr,tabl,new

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

1,1

{T(n,k)*k, k=1..n} setminus {0} = divisors of n; sum(T(n,k)*(k^i),k=1..n) = sigma[i](n) = sum of the i-th power of positive divisors of n; sum(T(n,k),k=1..n)=A000005, sum(T(n,k)*k,k=1..n)=A000203

Row sums are A000005. Diagonal sums are A032741(n+2). Might be called a Mobius matrix. Binomial transform (product by binomial matrix) is A101508. - Paul Barry (pbarry(AT)wit.ie), Dec 05 2004

A054525 = the inverse of this triangle = A129360 * A115369. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 15 2007

If the 1 in the lower right corner is moved to the upper right corner then the determinant gives the mobius function. [From Mats Granvik (mats.granvik(AT)abo.fi), Nov 18 2008]

Mats Granvik, Illustration of A051731

Jeffrey Ventrella, Divisor Plot [From Mats Granvik (mats.granvik(AT)abo.fi), Feb 08 2009]

T(n, k)=T(n-k, k) for k<=n/2, T(n, k)=0 for n/2<k<=n-1, T(n, n)=1

Rows given by A074854 converted to binary. Example: A074854(4)= 13(decimal)= 1101(binary); row 4 = 1, 1, 0, 1. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Oct 04 2003

Columns have g.f. x^k/(1-x^(k+1)) (k>=0). - Paul Barry (pbarry(AT)wit.ie), Dec 05 2004

Matrix inverse of triangle A054525, where A054525(n, k) = MoebiusMu(n/k) if k|n, 0 otherwise. - Paul D. Hanna (pauldhanna(AT)juno.com), Jan 09 2006

Equals = A129372 * A115361 as infinite lower triangular matrices. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 15 2007

This triangle * [1,2,3,...] = Sigma(n), A000203: (1, 3, 4, 7, 6, 12, 8,...). A051731 * [1/1, 1/2, 1/3,...] = Sigma(n)/n: (1/1, 3/2, 4/3, 7/4, 6/5,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 10 2007

T(n,k) = 0^(n mod k). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 01 2009]

T(n,k) = (1-((-1)^A172119))/2. [From Mats Granvik (mats.granvik(AT)abo.fi), Jan 26 2010]

Triangle begins:

.{1};

.{1,1};

.{1,0,1};

.{1,1,0,1};

.{1,0,0,0,1}; ...

Cf. A000005, A000203, A074854, A054525, A129372, A115361.

A077049 and A077051 are other presentations of this matrix. [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Apr 08 2009]

T(n,k) = A000007(A048158(n,k)). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 01 2009]

Sequence in context: A054524 A110471 A103994 this_sequence A135839 A155076 A120529

Adjacent sequences: A051728 A051729 A051730 this_sequence A051732 A051733 A051734

easy,nice,nonn,tabl,new

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)

0,1

With offset 2, this is the second bit in the binary expansion of n. [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Feb 13 2009]

R. Stephan, Some divide-and-conquer sequences ...

R. Stephan, Table of generating functions

a(n) = floor(log[2](4*(n+2)/3)) - floor(log[2](n+2)). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 22 2003

For n >= 2, a(n-2)=1+floor(log[2](n/3))-floor(log[2](n/2)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 03 2003

G.f.: 1/x^2/(1-x) * (1/x + sum(k>=0, x^(3*2^k)-x^2^(k+1))). - Ralf Stephan, Jun 04 2003

Cf. A086694, A079882, A079945.

Sequence in context: A104894 A168393 A071986 this_sequence A059652 A108736 A079813

Adjacent sequences: A079941 A079942 A079943 this_sequence A079945 A079946 A079947

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com), Feb 21 2003

1,8

If k > n/2, T(n,k) = P(n-k) = A000041(n-k). - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jan 12 2006

A002865(n) = Sum(a(n-k+1,k-1): 1<k<=floor((n+2)/2). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 04 2007

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 831.

L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 94, 96 and 307.

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 219.

D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIV.2, p. 493.

Franklin T. Adams-Watters, First 100 rows, flattened

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

H. Bottomley, Illustration of initial terms

D. J. Broadhurst and D. Kreimer, Towards cohomology of renormalization...

W. Lang, First 10 rows and more.

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

T(n, k)=Sum{T(n-k, i)}, 1<=i<=k for 1<=k<=n-1; T(n, n)=1 for n >= 1.

Or, T(n, 1) = T(n, n) = 1, T(n, k) = 0 (k>n), T(n, k) = T(n-1, k-1) + T(n-k, k).

G.f. for k-th column: x^k/(product(1-x^j, j=1..k)) - Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 29 2000

G.f.: A(x, y) = Product_{n>=1} 1/(1-x^n)^(P_n(y)/n), where P_n(y) = Sum_{d|n} eulerphi(n/d)*y^d. - Paul D. Hanna (pauldhanna(AT)juno.com), Jul 13 2004

G.f.=G(t,x)=-1+1/product(1-tx^j,j=1..infinity). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 12 2006

Triangle begins:

1;

1,1;

1,1,1;

1,2,1,1;

1,2,2,1,1;

1,3,3,2,1,1; ...

T(7,3)=4 because we have [3,3,1], [3,2,2], [3,2,1,1] and [3,1,1,1,1], each having greatest part 3; or [5,1,1], [4,2,1], [3,3,1] and [3,2,2] each having 3 parts.

G:=-1+1/product(1-t*x^j, j=1..15): Gser:=simplify(series(G, x=0, 17)): for n from 1 to 14 do P[n]:=coeff(Gser, x^n) od: for n from 1 to 14 do seq(coeff(P[n], t^j), j=1..n) od; # yields sequence in triangular form - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 12 2006

with(combstruct):for n from 0 to 18 do seq(count(Partition(n), size=m) , m = 1 .. n) od; # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 30 2009]

Cf. A000041 (row sums), A038497, A038498, A039805-A039809, A060016. Read from right to left gives A058398. Partial sums of rows gives A026820.

Column 3 is A001399.

First difference triangle of triangle A026820.

Sequence in context: A137350 A166240 A114087 this_sequence A114088 A037306 A007424

Adjacent sequences: A008281 A008282 A008283 this_sequence A008285 A008286 A008287

nonn,tabl,nice,easy

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

0,11

a(n)=floor[log10(10n)] = A004216(n)+1 =A004218(n+1)

Join[{1}, Array[ Floor[ Log[10, 10# ]] &, 104]] (from Robert G. Wilson v (rgwv(at)rgwv.com), Jan 04 2006)

(PARI) A055642(n)=#Str(n) [From M. F. Hasler (MHasler(AT)univ-ag.fr), Nov 17 2008]

Sequence in context: A036453 A040000 A007395 this_sequence A138902 A036452 A102572

Adjacent sequences: A055639 A055640 A055641 this_sequence A055643 A055644 A055645

base,easy,nonn,nice

Henry Bottomley (se16(AT)btinternet.com), Jun 06 2000

0,6

T(n,k) = number of integer strings s(0),...,s(n) such that s(0)=0, s(n)=k, s(i)=s(i-1)+c, where c is 0, 1 or 2. Columns of T include A002426, A005717 and A014531.

Also number of ordered trees having n+1 leaves, all at level three and n+k+3 edges. Example: T(3,5)=3 because we have three ordered trees with 4 leaves, all at level three and 11 edges: the root r has three children; from one of these children two paths of length two are hanging (i.e. 3 possibilities) while from each of the other two children one path of length two is hanging. Diagonal sums are the tribonacci numbers; more precisely: Sum(T(n-i,i), i=0..floor(2n/3)) = A000073(n+2). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 03 2004

T(n,k) = A111808(n,k) for 0<=k<=n and T(n,2*n-k) = A111808(n,k) for 0<=k<n. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 17 2005

The trinomial coefficients, T(n,i), are the absolute value of the coefficients of the chromatic polynomial of P_2 X P_n factored with x(x-1)^i terms. Example: The chromatic polynomial of P_2xP_2 is: x(x-1) - 2x(x-1)^2 + x(x-1)^3 and so T(1,0)=1, T(1,1)=2 and T(1,1)=1. - Thomas J Pfaff (tpfaff(AT)ithaca.edu), Oct 02 2006

T(n,k) is the number of distinct ways in which k unlabeled objects can be distributed in n labeled urns allowing at most 2 objects to fall in each urn. - Nour-Eddine Fahssi (fahssin(AT)yahoo.fr), Mar 16 2008

F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999) 73-112.

B. A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8. English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 17.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.

D. C. Fielder and C. O. Alford, Pascal's triangle: top gun or just one of the gang?, in G E Bergum et al., eds., Applications of Fibonacci Numbers Vol. 4 1991 pp. 77-90 (Kluwer).

A. Fink, R. K. Guy and M. Krusemeyer, Partitions with parts occurring at most thrice, in preparation.

V. E. Hoggatt, Jr. and M. Bicknell, Diagonal sums of generalized Pascal triangles, Fib. Quart., 7 (1969), 341-358, 393.

L. Kleinrock, Uniform permutation of sequences, JPL Space Programs Summary, Vol. 37-64-III, Apr 30, 1970, pp. 32-43.

L. W. Shapiro, S. Getu, W.-J. Woan and L. C. Woodson, The Riordan group, Discrete Applied Math., 34 (1991), 229-239.

G. E. Andrews, "Euler's `exemplum memorabile inductionis fallacis' and q-trinomial coefficients", J. Amer. Math. Soc. 3 (1990) 653-669.

Freund, J. E., Restricted Occupancy Theory - A Generalization of Pascal's Triangle, American Mathematical Monthly, Vol. 63, No. 1 (1956), pp. 20-27.

T. D. Noe, Rows n=0..30 of triangle, flattened

S. R. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle (arXiv:0802.2654)

G. E. Andrews, Three aspects of partitions

L. Euler, On the expansion of the power of any polynomial (1+x+x^2+x^3+x^4+etc)^n

L. Euler, De evolutione potestatis polynomialis cuiuscunque (1+x+x^2+x^3+x^4+etc)^n E709

W. Florek and T. Lulek, Combinatorial analysis of magnetic configurations

S. Kak, The Golden Mean and the Physics of Aesthetics

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

Eric Weisstein's World of Mathematics, Trinomial Coefficient

G.f.: 1/(1-z(1+w+w^2)). T(n, k) = Sum_{0 <= r <= k/3} (-1)^r*C(n, r)*C(k-3*r+n-1, n-1).

T(i, 0) = T(i, 2i) = 1 for i >= 0, T(i, 1) = T(i, 2i-1) = i for i >= 1 and for i >= 2 and 2 <= j <= i-2, T(i, j) = T(i-1, j-2)+T(i-1, j-1)+T(i-1, j).

The row sums are powers of 3 (A000244). - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Aug 14 2004

T(n, k) = Sum{i=0..[k/2], C(n, 2i+n-k)*C(2i+n-k, i) }. - R. Stephan, Jan 26 2005

T(n, k):=sum{j=0..n, C(n, j)C(j, k-j)} - Paul Barry (pbarry(AT)wit.ie), May 21 2005

T(n, k)=sum{j=0..n, C(k-j, j)C(n, k-j)}; - Paul Barry (pbarry(AT)wit.ie), Nov 04 2005

T(n,k)=sum{j=0..n,(-1)^j*C(n,j)*C(2n-2j,k-j)};(G. E. Andrews (1990)); obtained by expanding [(1+x)^2-x]^n. - Loic Turban (turban(AT)lpm.u-nancy.fr), Aug 31 2006

T(n,k)=sum{j=0..n,C(n,j)*C(n-j,k-2j)}; obtained by expanding [(1+x)+x^2]^n. - Loic Turban (turban(AT)lpm.u-nancy.fr), Aug 31 2006

T(n,k)=(-1)^k*sum{j=0..n,(-3)^j*C(n,j)*C(2n-2j,k-j)} (a); obtained by expanding [(1-x)^2+3x]^n. - Loic Turban (turban(AT)lpm.u-nancy.fr), Aug 31 2006

T(n,k)=(1/2)^k*sum{j=0..n,3^j*C(n,j)*C(2n-2j,k-2j)} (b); obtained by expanding [(1+x/2)^2+(3/4)*x^2]^n. - Loic Turban (turban(AT)lpm.u-nancy.fr), Aug 31 2006

T(n,k)=(2^k/4^n)*sum{j=0..n,3^j*C(n,j)*C(2n-2j,k)} (c); obtained by expanding [(1/2+x)^2+3/4]^n; follows from (c) using T(n,k)=T(2n-k). - Loic Turban (turban(AT)lpm.u-nancy.fr), Aug 31 2006

1; 1,1,1; 1,2,3,2,1; 1,3,6,7,6,3,1; ...

# To get n-th row: expand((1+x+x^2)^n);

(PARI) T(n, k)=if(n<0, 0, polcoeff((1+x+x^2)^n, k))

Columns of T include A002426, A005717, A014531, A005581, A005712 etc. See also A035000, A008287.

Cf. A000073.

First differences are in A025177. Pairwise sums are in A025564.

Cf. A123531.

Sequence in context: A092542 A026552 A086437 this_sequence A026323 A017838 A058294

Adjacent sequences: A027904 A027905 A027906 this_sequence A027908 A027909 A027910

nonn,tabf,nice

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

0,1

Also triangle giving successive states of cellular automaton generated by "Rule 60" and "Rule 102". - Hans Havermann (pxp(AT)rogers.com), May 26 2002

Also triangle formed by reading triangle of Eulerian numbers (A08292) mod 2. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Oct 02 2003

Self-inverse when regarded as an infinite lower triangular matrix over GF(2).

Start with [1], repeatedly apply the map 0 -> [00/00], 1 -> [10/11] [Allouche and Berthe]

Also triangle formed by reading triangles A011117, A028338, A039757, A059438, A085881, A086646, A086872, A087903, A0104219 mod 2 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 18 2005

J. H. Conway writes (in Math Forum): at least the first 31 rows give odd-sided constructible polygons (sides 1, 3, 5, 15,17 ... see A001317). The 1's form a Sierpinski sieve. - M. Dauchez (mdzzdm(AT)yahoo.fr), Sep 19 2005

When regarded as an infinite lower triangular matrix, its inverse is a (-1,0,1)-matrix with zeros undisturbed and the nonzero entries in every column form the Prouhet-Thue-Morse sequence (1,-1,-1,1,-1,1,1,-1,...) A010060 (up to relabeling). - David Callan (callan(AT)stat.wisc.edu), Oct 27 2006

Triangle read by rows: antidiagonals of an array formed by successive iterates of running sums mod 2, beginning with (1, 1, 1,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 10 2008

R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.

Y. Moshe, The distribution of elements in automatic double sequences, Discr. Math., 297 (2005), 91-103.

H.-O. Peitgen, H. Juergens and D. Saupe: Chaos and Fractals (Springer-Verlag 1992), p. 408.

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; Chapter 3.

T. D. Noe, Rows n=0..50 of triangle, flattened

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

J.-P. Allouche and V. Berthe, Triangle de Pascal, complexite et automates.

J. Baer, Explore patterns in Pascal's Triangle

A. Bogomolny, Dot Patterns and Sierpinski Gasket

S. Butkevich, Pascal Triangle Applet

B. Cherowitzo, Pascal's Triangle using Clock Arithmetic, Part I

B. Cherowitzo, Pascal's Triangle using Clock Arithmetic, Part II

A. Granville, Pascal's Triangle Interface

I. Kobayashi et al., Pascal's Triangle

Dr. Math, Regular polygon formulas>Regular polygon formulas

National Curve Bank, Sierpinski Triangles

F. Richman, Pascal's triangle modulo n

F. Richman, Pascal's triangle modulo n

Eric Weisstein's World of Mathematics, Sierpinski Sieve

Eric Weisstein's World of Mathematics, Rule 60

Eric Weisstein's World of Mathematics, Rule 102

Index entries for sequences related to cellular automata

Index entries for triangles and arrays related to Pascal's triangle

Index entries for sequences generated by sieves

Sum_{k>=0} T(n, k) = A001316(n) = 2^A000120(n).

T(n,k) = T(n-1,k-1) XOR T(n-1,k), 0<k<n; T(n,0) = T(n,n) = 1. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 13 2009]

Triangle begins:

.1,

.1,1,

.1,0,1,

.1,1,1,1,

.1,0,0,0,1,

.1,1,0,0,1,1,

.1,0,1,0,1,0,1,

.1,1,1,1,1,1,1,1,

.1,0,0,0,0,0,0,0,1,

.1,1,0,0,0,0,0,0,1,1,

.1,0,1,0,0,0,0,0,1,0,1,

.1,1,1,1,0,0,0,0,1,1,1,1,

.1,0,0,0,1,0,0,0,1,0,0,0,1,

....

Mod[ Flatten[ NestList[ Prepend[ #, 0] + Append[ #, 0] &, {1}, 13]], 2] (from Robert G. Wilson v May 26 2004)

Contribution from Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Oct 10 2009: (Start)

(PARI) \\ Recurrence for Pascal's triangle mod p, here p = 2.

p = 2; s=13; T=matrix(s, s); T[1, 1]=1;

for(n=2, s, T[n, 1]=1; for(k=2, n, T[n, k] = (T[n-1, k-1] + T[n-1, k])%p ));

for(n=1, s, for(k=1, n, print1(T[n, k], ", "))) (End)

Cf. A007318, A054431, A001317, A008292, A083093, A034931, A034930, A008975, A034932.

Sequence in context: A144093 A143200 A166282 this_sequence A054431 A164381 A106470

Adjacent sequences: A047996 A047997 A047998 this_sequence A048000 A048001 A048002

nonn,tabl,easy,nice

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

Additional links from Lekraj Beedassy (blekraj(AT)yahoo.com), Jan 22 2004

0,3

A. Cobham, Uniform Tag Sequences, Mathematical Systems Theory, 6 (1972), 164-192.

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

David W. Wilson, Table of n, a(n) for n = 0..10000

a(n) = [n / 10^([log_10(n)])] where [] denotes floor and log_10(n) is the logarithm is base 10. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 07 2001

23 begins with a 2, so a(23) = 2.

a[n_] := First[IntegerDigits[n]]

(PARI) a(n)=if(n<1, 0, if(n<10, n, a(n\10)))

Cf. A061681, A130571, A109453, A134010.

A002993, A089951, A002994, A143464, A098174, A098175, A072543, A072544, A073600, A073601, A037904. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 17 2008]

Sequence in context: A052423 A126616 A121042 this_sequence A134777 A004427 A113230

Adjacent sequences: A000027 A000028 A000029 this_sequence A000031 A000032 A000033

nonn,base,easy,nice

N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)

1,12

Comments following discussions with David Applegate (david(AT)research.att.com), May 05, 2006: Certainly a(10) = -1 and probably a(n) is always -1 if n is a multiple of 10. Furthermore a(15) is almost certainly -1: T_15 has not reached a cycle in 10^7 terms (see A118532).

Comment from Martin Fuller, May 12 2006: If n is a multiple of 10 the operation can never generate a trailing zero and so is reversible. So it loops only if it returns to the start, which is impossible. Hence a(10k) = -1.

Comment from Martin Fuller, May 12 2006: I suspect a(115) = 385592406, A117817(115) = 79560. Can someone confirm?

The -1 entries for n >= 0 both here and in A117817 are presently only conjectural.

N. J. A. Sloane, Sequences of RADD type

T_2 enters a cycle of length 81 after 1 step.

ReverseNum[n_] := FromDigits[Reverse[IntegerDigits[n]]]; maxLen=10000; Table[z=1; lst={1}; While[z=ReverseNum[z]+n; !MemberQ[lst, z] && Length[lst]<maxLen, AppendTo[lst, z]]; If[Length[lst]<maxLen, Position[lst, z][[1, 1]]-1, -1], {n, 100}] (Noe)

For T_1, T_2, ..., T_16 (omitting T_9, which is uninteresting) see A117230, A117521, A118517, A117828, A117800, A118525, A118526, A118527, A117841, A118528, A118529, A118530, A118531, A118532, A118533.

Cf. A117817.

Sequence in context: A062008 A091776 A069460 this_sequence A099189 A053234 A020896

Adjacent sequences: A117813 A117814 A117815 this_sequence A117817 A117818 A117819

sign,base

N. J. A. Sloane (njas(AT)research.att.com), following discussions with Luc Stevens, May 04 2006

a(21)-a(33) from Luc Stevens, May 08 2006

a(33) onwards from T. D. Noe (noe(AT)sspectra.com), May 10 2006

Further terms from Martin Fuller, May 12 2006

0,6

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 810.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.

H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 230.

G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.

H. H. Goldstine, A History of Numerical Analysis, Springer-Verlag, 1977; Section 2.6.

F. Lemmermeyer, Reciprocity Laws From Euler to Eisenstein, Springer-Verlag, 2000, p. 330.

H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1.

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

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

Zhi-Hong Sun, Congruences involving Bernoulli polynomials, Discr. Math., 308 (2007), 71-112.

S. Plouffe, Table of n, a(n) for n = 0..249 [taken from link below]

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

J. L. Arregui, Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles.

J. Butcher, Some applications of Bernoulli numbers

C. K. Caldwell, The Prime Glossary, Bernoulli number

R. Jovanovic, Bernoulli numbers and the Pascal triangle

M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.

B. C. Kellner, On irregular prime power divisors of the Bernoulli numbers

B. C. Kellner, The structure of Bernoulli numbers

C. Lin and L. Zhipeng, On Bernoulli numbers and its properties

S. O. S. Math, Bernoulli and Euler Numbers

Hisanori Mishima, Factorizations of many number sequences

Hisanori Mishima, Factorizations of many number sequences

Niels Nielsen, Traite Elementaire des Nombres de Bernoulli, Gauthier-Villars, 1923, pp. 398.

S. Plouffe, The 250,000-th Bernoulli Number

S. Plouffe, The First 498 Bernoulli numbers [Project Gutenberg Etext]

S. Ramanujan, Some Properties of Bernoulli's Numbers

S. S. Wagstaff, Prime factors of the absolute values of Bernoulli numerators

Eric Weisstein's World of Mathematics, More information.

Wikipedia, Bernoulli number

Index entries for sequences related to Bernoulli numbers.

E.g.f: t/(e^t - 1).

B_{2n}/(2n)! = 2*(-1)^(n-1)*(2*Pi)^(-2n) Sum_{k=1..inf} 1/k^(2n) (gives asymptotics) - Rademacher, p. 16, Eq. (9.1). In particular, B_{2*n} ~ (-1)^(n-1)*2*(2*n)!/(2*Pi)^(2*n).

B_{2n} = [ 1, 1/6, -1/30, 1/42, -1/30, 5/66, -691/2730, 7/6, -3617/510,... ].

bernoulli(n);

(PARI) a(n)=numerator(bernfrac(2*n))

B_n gives A027641/A027642. See A027641 for full list of references, links, formulae, etc.

See A002445 for denominators.

Cf. also A002882, A003245, A127187, A127188.

Sequence in context: A117709 A133750 A090947 this_sequence A092133 A071772 A157281

Adjacent sequences: A000364 A000365 A000366 this_sequence A000368 A000369 A000370

sign,frac,nice

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