A168259 - OEIS

%I #19 Nov 19 2022 14:51:34

%S 1,2,6,14,38,96,254,656,1724,4492,11776,30774,80608,210892,552226,

%T 1445374,3784308,9906482,25936206,67899344,177764618,465387226,

%U 1218404344,3189806746,8351034954,21863248282,57238759726,149852900454,392320072078,1027106974446,2689001192594,7039895709776

%N Eigensequence of triangle A168258.

%C 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.

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

%H 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.

%o (PARI) T(n, k) = if(binomial(k, n-k)>0, 1, 0); \\ A101688

%o 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

%o 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;

%o 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

%Y Cf. A104457, A168258, A168260.

%K eigen,nonn

%O 1,2

%A _Gary W. Adamson_, Nov 21 2009

%E Edited and more terms from _Michel Marcus_, Nov 19 2022