A168259 -id:A168259 - cp's OEIS frontend

A168260 Triangle read by rows, A168258 * the diagonalized variant of A168259.

1, 1, 1, 2, 2, 2, 2, 2, 4, 6, 3, 3, 6, 12, 14, 3, 3, 6, 18, 28, 38, 4, 4, 8, 24, 42, 76, 96, 4, 4, 8, 24, 56, 114, 192, 254, 5, 5, 10, 30, 70, 152, 288, 508, 656, 5, 5, 10, 30, 70, 190, 384, 762, 1312, 1724, 6, 6, 12, 36, 84, 228, 480, 1016

Offset: 1

Gary W. Adamson, Nov 21 2009

Row sums = A168259: (1, 2, 6, 14, 38, 96, ...).
Sum of n-th row terms = rightmost term of next row.
Conjecture: Row sum ratios tend to phi^2 = 2.6180339... (cf. A168259).

			Triangle begins:
  1;
  1, 1;
  2, 2,  2;
  2, 2,  4,  6;
  3, 3,  6, 12,  14;
  3, 3,  6, 18,  28,  38;
  4, 4,  8, 24,  42,  76,  96;
  4, 4,  8, 24,  56, 114, 192,  254;
  5, 5, 10, 30,  70, 152, 288,  508,  656;
  5, 5, 10, 30,  70, 190, 384,  762, 1312,  1724;
  6, 6, 12, 36,  84, 228, 480, 1016, 1968,  3448,  4492;
  6, 6, 12, 36,  84, 228, 576, 1270, 2624,  5172,  8984, 11776;
  7, 7, 14, 42,  98, 266, 672, 1524, 3284,  6896, 13476, 23552,  30774;
  7, 7, 14, 42,  98, 266, 672, 1778, 3936,  8620, 17968, 35328,  61548,  80608;
  8, 8, 16, 48, 112, 304, 768, 2032, 5248, 12068, 26952, 58880, 123096, 241824;
  ...

Cf. A168258, A168259.

Let M = triangle A168258 and Q = the diagonalized variant of M's eigensequence
such that Q's rightmost diagonal = A168259 prefaced with a 1: (1, 1, 2, 6, ...).
and other terms = 0.
Triangle A168260 = M * Q as infinite lower triangular matrices.

A168258 Triangle read by rows, A101688 * A000012 as infinite lower triangular matrices.

Original entry on oeis.org

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, 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, 6, 6, 6, 6, 6, 5, 4, 3, 2, 1

Offset: 1

Gary W. Adamson, Nov 21 2009

Row sums = A001318, general pentagonal numbers: (1, 2, 5, 12, 15, 22, ...).
Eigensequence of the triangle = A168259: (1, 2, 6, 14, 38, 96, 254, 656, ...).
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

			First few rows of the triangle:
  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, 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, 6, 6, 6, 6, 6, 5, 4, 3, 2, 1;
  7, 7, 7, 7, 7, 7, 7, 6, 5, 4, 3, 2, 1;
  7, 7, 7, 7, 7, 7, 7, 7, 6, 5, 4, 3, 2, 1;
  8, 8, 8, 8, 8, 8, 8, 8, 7, 6, 5, 4, 3, 2, 1;
  ...

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

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

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

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

Name corrected by Gary W. Adamson, Nov 15 2022

cp's OEIS Frontend

A168260 Triangle read by rows, A168258 * the diagonalized variant of A168259.

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