A168258 - OEIS

%I #23 Nov 17 2022 04:47:31

%S 1,1,1,2,2,1,2,2,2,1,3,3,3,2,1,3,3,3,3,2,1,4,4,4,4,3,2,1,4,4,4,4,4,3,

%T 2,1,5,5,5,5,5,4,3,2,1,5,5,5,5,5,5,4,3,2,1,6,6,6,6,6,6,5,4,3,2,1,6,6,

%U 6,6,6,6,6,5,4,3,2,1

%N Triangle read by rows, A101688 * A000012 as infinite lower triangular matrices.

%C Row sums = A001318, general pentagonal numbers: (1, 2, 5, 12, 15, 22, ...).

%C Eigensequence of the triangle = A168259: (1, 2, 6, 14, 38, 96, 254, 656, ...).

%C The operation A101688 * A000012 transforms rows of A101688 into sequence terms by taking partial sums from the right of A101688 rows. For example, row 3 of A101688 (0, 0, 1, 1) becomes (2, 2, 2, 1). - _Gary W. Adamson_, Nov 15 2022

%F Triangle read by rows, A101688 * A000012 as infinite lower triangular matrices.

%F a(n) = min(A004736, A204164); a(n) = min(j, floor((t+2)/2)), where j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). - _Boris Putievskiy_, Apr 18 2013

%e First few rows of the triangle:

%e   1;

%e   1, 1;

%e   2, 2, 1;

%e   2, 2, 2, 1;

%e   3, 3, 3, 2, 1;

%e   3, 3, 3, 3, 2, 1;

%e   4, 4, 4, 4, 3, 2, 1;

%e   4, 4, 4, 4, 4, 3, 2, 1;

%e   5, 5, 5, 5, 5, 4, 3, 2, 1;

%e   5, 5, 5, 5, 5, 5, 4, 3, 2, 1;

%e   6, 6, 6, 6, 6, 6, 5, 4, 3, 2, 1;

%e   6, 6, 6, 6, 6, 6, 6, 5, 4, 3, 2, 1;

%e   7, 7, 7, 7, 7, 7, 7, 6, 5, 4, 3, 2, 1;

%e   7, 7, 7, 7, 7, 7, 7, 7, 6, 5, 4, 3, 2, 1;

%e   8, 8, 8, 8, 8, 8, 8, 8, 7, 6, 5, 4, 3, 2, 1;

%e   ...

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

%o lista(nn) = my(ma=matrix(nn+1, nn, n, k, T(n-1, k-1)), mb=matrix(nn, nn, n, k, n>=k)); my(m=ma*mb, list=List()); for (n=1, nn, listput(list, vector(n, k, m[n,k]))); Vec(list); \\ _Michel Marcus_, Nov 16 2022

%Y Cf. A001318, A101688, A000012, A168259, A004736, A204164.

%K nonn,tabl

%O 1,4

%A _Gary W. Adamson_, Nov 21 2009

%E Name corrected by _Gary W. Adamson_, Nov 15 2022