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proposed

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.

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.

More Edited and more terms from Michel Marcus, Nov 19 2022

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

Triangle A168258 * the diagonalized variant of A168259 = triangle A168270 having row sums = A168259.

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

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} QM^n, resulting in a left-shifted vector. Delete the first 1, getting (1, 2, 6, 14, 38, 96, ...) = this sequence. . Q^n} Q tri.

(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

Cf. A104457, A168258, A168260.

eigen,nonn,more,changed

More terms from Michel Marcus, Nov 19 2022

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proposed

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_lim_{n->oo} Q^n; , resulting in a left-shifted vector. Delete the first 1, getting (1, 2, 6, 14, 38, 96, ...) = this sequence. . Q^n} Q tri.

proposed

editing

editing

proposed

Eigensequence of triangle A168258. Triangle A168258 * the diagonalized variant of A168259 = triangle A168270 having row sums = A168259.

Eigensequence of triangle A168258.

Triangle A168258 * the diagonalized variant of A168259 = triangle A168270 having row sums = A168259.

P. Paul Barry, <a href="http://arxiv.org/abs/1107.5490">Invariant number triangles, eigentriangles and Somos-4 sequences</a>, arXiv preprint arXiv:1107.5490, [math.CO], 2011.

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->inf.oo} Q^n; resulting in a left-shifted vector. Delete the first 1, getting (1, 2, 6, 14, 38, 96,...) = A168259this sequence. . Q^n} Q tri.

Cf. A168258, A168260.

eigen,nonn,more

approved

editing

P. Barry, <a href="http://arxiv.org/abs/1107.5490">Invariant number triangles, eigentriangles and Somos-4 sequences</a>, Arxiv arXiv preprint arXiv:1107.5490, 2011