A168260 -id:A168260 - cp's OEIS frontend

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

Gary W. Adamson, Nov 21 2009

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...

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

Cf. A104457, A168258, A168260.

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

Edited and more terms from Michel Marcus, Nov 19 2022

cp's OEIS Frontend

A168259 Eigensequence of triangle A168258.

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