A168317 Eigensequence of triangle A168316.

1, 1, 1, 3, 6, 16, 39, 103, 263, 690, 1791, 4693, 12247, 32073, 83869, 219598, 574658, 1504540, 3938272, 10310703, 26992036, 70666516, 185002819, 484344843, 1268019438, 3319721050, 8691111639, 22753633693, 59569705507, 155955534688, 408296678959, 1068934637918, 2798506659973

Offset

0,4

Comments

Conjectured convergent of a(n)/a(n-1) = phi^2 = 2.6180339... E.g.: a(19)/a(18) = 10310703/3938272 = 2.6180779....

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Formula

Equals lim_{n->oo} M^2, the left shifted vector considered as a sequence where M = triangle A168316 as an infinite lower triangular matrix, then shifted down one row, and inserting a "1" at top.

Prog

(PARI) EigenSeq(nn, tf)={my(v=vector(nn+1)); v[1] = 1; for(n=2, #v, v[n] = sum(k=0, n-2, tf(n-2, k)*v[k+1]); ); v}
{ my(N=32, M=matrix(N, N, n, k, k>n\2 && k<=n)^2); EigenSeq(N, (n, k)->M[1+n, 1+k]) } \\ Andrew Howroyd, Sep 23 2025

Crossrefs

Cf. A104457, A168316, A168318.

Sequence in context: A279685 A001675 A248091 * A188442 A046211 A239980

Adjacent sequences: A168314 A168315 A168316 * A168318 A168319 A168320

Keyword

eigen,nonn

Author

Gary W. Adamson, Nov 22 2009

Extensions

a(0)=1 prepended and a(20) onwards from Andrew Howroyd, Sep 23 2025

Status

approved