login
A168259
Eigensequence of triangle A168258.
2
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
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
KEYWORD
eigen,nonn
AUTHOR
Gary W. Adamson, Nov 21 2009
EXTENSIONS
Edited and more terms from Michel Marcus, Nov 19 2022
STATUS
approved