0,3

Denominators of continued fraction convergents to (3+Sqrt[13])/2. - Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 14 2003

A006190(n) and A006497(n) occur in pairs: (a,b): (1,3), (3,11), (10,36), (33,119), (109,393)...such that b^2 - 13a^2 = 4(-1)^n. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 15 2003

Form the 4-node graph with matrix A=[1,1,1,1;1,1,1,0;1,1,0,1;1,0,1,1]. Then A006190 counts the walks of length n from the vertex with degree 5 to one (any) of the other vertices. - Paul Barry (pbarry(AT)wit.ie), Oct 02 2004

a(n+1) is the binomial transform of A006138. - Paul Barry (pbarry(AT)wit.ie), May 21 2006

a(n+1) is the diagonal sum of the exponential Riordan array (exp(3x),x). - Paul Barry (pbarry(AT)wit.ie), Jun 03 2006

Number of paths in the right half-plane from (0,0) to the line x=n-1, consisting of steps U=(1,1), D=(1,-1), h=(1,0), and H=(2,0). Example: a(3)=10 because we have hh, H, UD, DU, hU, Uh, UU, hD, Dh, and DD. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 03 2007

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

H. L. Abbott and D. Hanson, A lattice path problem, Ars Combin., 6 (1978), 163-178.

A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 128.

A. F. Horadam, Generating identities for generalized Fibonacci and Lucas triples, Fib. Quart., 15 (1977), 289-292.

Ayoub B. Ayoub, "Fibonacci-like sequences and Pell equations", The College Mathematics Journal, Vol. 38 (2007), pp. 49-53.

W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, A Pascal rhombus, Fibonacci Quarterly, 35 (1997), 318-328.

T. D. Noe, Table of n, a(n) for n=0..200

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

Tanya Khovanova, Recursive Sequences

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 158

Index entries for sequences related to Chebyshev polynomials.

G.f.: x/(1-3*x-x^2).

a(3n)=2*A041019(5n-1), a(3n+1)=A041019(5n), a(3n+1)=A041019(5n+1).

n>=1 a(2n)=3*A004190(n-1); a(3n)=10*A041613(n-1) - Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 14 2003

a(n-1) + a(n+1) = A006497(n); [A006497(n)]^2 - 13[a(n)]^2 = 4(-1)^n - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 15 2003

a(n)=U(n-1, (3/2)i)(-i)^(n-1), i^2=-1 - Paul Barry (pbarry(AT)wit.ie), Nov 19 2003

a(n)=sum{k=0..n-1, binomial(n-k-1, k)3^(n-2k-1)} - Paul Barry (pbarry(AT)wit.ie), Oct 02 2004

a(n)=F(n, 3), the n-th Fibonacci polynomial evaluated at x=3.

Let M = {{0, 1}, {1, 3}}, v[1] = {0, 1}, v[n] = M.v[n - 1]; then a(n) = Abs[v[n][[1]]]. - Roger L. Bagula (rlbagulatftn(AT)yahoo.com), May 29 2005

a(n+1)=sum{k=0..n, sum{j=0..n-k, C(k,j)C(n-j,k)*2^(k-j)}}=sum{k=0..n, sum{j=0..n-k, C(k,j)C(n-j,k)*2^(n-j-k)}}; a(n+1)=sum{k=0..floor(n/2), comb(n-k,k)3^(n-2k)}=sum{k=0..n, comb(k,n-k)3^(2k-n)}; - Paul Barry (pbarry(AT)wit.ie), May 21 2006

E.g.f.: exp(3x/2)*sinh(sqrt(13)x/2)/(sqrt(13)/2); - Paul Barry (pbarry(AT)wit.ie), Jun 03 2006

a(n)=(ap^n-am^n)/(ap-am), with ap := (3+sqrt(13))/2, am := (3-sqrt(13))/2.

Let C = (3+sqrt(13))/2 = exp ArcSinh(3/2) = 3.3027756377...Then C^n, n>0 = a(n)*(1/C) + a(n+1). Example: C^3 = 36.02775637... = a(3)*(1/C) + a(4) = 10*(.302775637...) + 33. Let X = the 2 X 2 matrix [0, 1; 1, 3]. Then X^n = [a(n-1), a(n); a(n), a(n+1)]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 21 2007

1/3 = 3/(1*10) + 3/(3*33) + 3/(10*109) + 3/(33*360) + 3/(109*1189) + ... - Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 16 2008

a[0]:=0: a[1]:=1: for n from 2 to 25 do a[n]:= 3*a[n-1]+a[n-2] end do: seq(a[n], n=0..25); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 03 2007

A006190:=-1/(-1+3*z+z**2); [Conjectured by S. Plouffe in his 1992 dissertation.]

with(combinat): a:=n->fibonacci(n, 3)-3*fibonacci(n-1, 3): seq(a(n), n=2..27); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 04 2008

a[n_] := (MatrixPower[{{1, 3}, {1, 2}}, n].{{1}, {1}})[[2, 1]]; Table[ a[n], {n, -1, 24}] (from Robert G. Wilson v Jan 13 2005)

(PARI) a(n)=if(n<1, 0, contfracpnqn(vector(n, i, 2+(i>1)))[2, 1])

Row sums of Pascal's rhombus (A059317). Also row sums of triangle A054456(n, m). Cf. A000045, A000129, A001076.

Cf. A006497, A052906.

Adjacent sequences: A006187 A006188 A006189 this_sequence A006191 A006192 A006193

Sequence in context: A126184 A060557 A018920 this_sequence A020704 A113299 A126931

easy,nonn,nice

njas

More terms from Larry Reeves (larryr(AT)acm.org), Apr 17 2000