0,3

Counts walks of length n at the vertex of degree five in the graph with adjacency matrix A=[1,1,1,1;1,0,0,0;1,0,0,0;1,0,0,0]. - Paul Barry (pbarry(AT)wit.ie), Oct 02 2004

Form the graph with matrix A=[0,1,1,1;1,1,0,0;1,0,1,0;1,0,0,1]. The sequence 0,1,1,4,... counts walks of length n between the vertex without loop and another vertex. - Paul Barry (pbarry(AT)wit.ie), Oct 02 2004

Hankel transform is the sequence [1,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,...] - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 10 2007

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

A. K. Whitford, Binet's formula generalized, Fib. Quart., 15 (1977), pp. 21, 24, 29.

Stephen Wolfram, 'The Mathematica Book,' Fourth Edition, Wolfram Media or Cambridge University Press, 1999, p. 96.

Joerg Arndt, Fxtbook

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 436

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.

a(n)={[ (1+sqrt(13))/2 ]^(n+1) - [ (1-sqrt(13))/2 ]^(n+1)}/sqrt(13).

a(n)=sum(k=0, ceil(n/2), 3^k*binomial(n-k, k)) - Benoit Cloitre - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 07 2004

a(0) = 1; a(1) = 1; for n>=1, a(n+1) = (a(n)^2 - (-3)^n) / a(n-1) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 07 2004

The i-th term of the sequence is the (1, 2) entry in the i-th power of the 2 by 2 matrix M=((-1, 1), (1, 2)). - Simone Severini (ss54(AT)york.ac.uk), Oct 15 2005

a(n) = lower right term in the 2 X 2 matrix [0,3; 1,1]^n - Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 02 2008

a(n)=Sum_{k, 0<=k<=n} A109466(n,k)*(-3)^(n-k). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 26 2008]

a(n)=Product[(1 + 12*Cos[k*Pi/n]^2), {k, 1, Floor[(n - 1)/2]}]. [From Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Nov 21 2008]

Limiting ratio = (1 + Sqrt[1 + 12])/2=2.302775637731995. [From Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Nov 21 2008]

a:=n->sum(binomial(n-k, k)*3^k, k=0..n): seq(a(n), n=0..29); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Sep 30 2006

A006130:=-1/(-1+z+3*z**2); [S. Plouffe in his 1992 dissertation.]

a[0] = a[1] = 1; a[n_] := a[n] = a[n - 1] + 3a[n - 2]; Table[ a[n], {n, 0, 30}]

f[n_] = Product[(1 + 12*Cos[k*Pi/n]^2), {k, 1, Floor[(n - 1)/2]}]; Table[FullSimplify[ExpandAll[f[n]]], {n, 0, 15}] N[%] [From Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Nov 21 2008]

sage: from sage.combinat.sloane_functions import recur_gen2 sage: it = recur_gen2(1, 1, 1, 3) sage: [it.next() for i in range(30)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008

(Other) sage: [lucas_number1(n, 1, -3) for n in xrange(1, 31)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]

Cf. A006131, A015440.

Cf. A052533.

Sequence in context: A102991 A062306 A140167 this_sequence A133264 A127415 A045548

Adjacent sequences: A006127 A006128 A006129 this_sequence A006131 A006132 A006133

nonn

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

Edited by Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 05, 2002