A168259 Eigensequence of triangle A168258.

1, 2, 6, 14, 38, 96, 254, 656, 1724, 4492, 11776, 30774, 80608, 210892, 552226, 1445374, 3784308, 9906482, 25936206, 67899344, 177764618, 465387226, 1218404344, 3189806746, 8351034954, 21863248282, 57238759726, 149852900454, 392320072078, 1027106974446, 2689001192594, 7039895709776

Offset

1,2

Comments

Eigensequence of triangle A168258, derived from the following operation: Shift down triangle A168258, so that rows begin [1; 1; 1,1; 2,2,1; ...] = triangle M. Then take lim_{n->oo} M^n, resulting in a left-shifted vector. Delete the first 1, getting (1, 2, 6, 14, 38, 96, ...) = this sequence.

a(n)/a(n-1) apparently tends to phi^2=A104457. a(19)/a(18) = 2.618104...

Links

Table of n, a(n) for n=1..32.

Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv preprint arXiv:1107.5490 [math.CO], 2011.

Prog

(PARI) T(n, k) = if(binomial(k, n-k)>0, 1, 0); \\ A101688
mat(nn) = my(ma=matrix(nn+1, nn, n, k, T(n-1, k-1)), mb=matrix(nn, nn, n, k, n>=k)); ma*mb; \\ A168258
shiftm(m, nn) = my(shm=matrix(nn+1, nn+1)); shm[1, 1]=1; for (n=1, nn, for(k=1, nn, shm[n+1, k] = m[n, k]; ); ); shm;
lista(nn) = my(m=mat(nn), shm=shiftm(m, nn), shmnn=shm^nn); vector(nn, k, shmnn[k+1, 1]); \\ Michel Marcus, Nov 19 2022

Crossrefs

Cf. A104457, A168258, A168260.

Sequence in context: A026597 A122112 A190788 * A275208 A000634 A006654

Adjacent sequences: A168256 A168257 A168258 * A168260 A168261 A168262

Keyword

eigen,nonn

Author

Gary W. Adamson, Nov 21 2009

Extensions

Edited and more terms from Michel Marcus, Nov 19 2022

Status

approved