A391318 Triangle array read by rows: T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-1) + 1 with T(0,0) = T(n,-1) = T(n,n+1) = 0.

0, 1, 1, 2, 3, 2, 3, 7, 7, 3, 4, 13, 18, 13, 4, 5, 21, 39, 39, 21, 5, 6, 31, 74, 97, 74, 31, 6, 7, 43, 127, 211, 211, 127, 43, 7, 8, 57, 202, 413, 520, 413, 202, 57, 8, 9, 73, 303, 743, 1145, 1145, 743, 303, 73, 9, 10, 91, 434, 1249, 2302, 2811, 2302, 1249, 434, 91, 10

Offset

0,4

Comments

Column k=1 appears to be A002061(n).

Column k=2 appears to be A051743(n-1) - A000332(n-2), starting from n=2.

Links

Leonidas Liponis, Table of n, a(n) for n = 0..5150 (100 rows flattened)

Formula

T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-1) + 1 with T(0,0) = T(n,-1) = T(n,n-k) = 0.

Example

Triangle begins:
  Sum | Terms
    0 |   0;
    2 |   1,   1;
    7 |   2,   3,   2;
   20 |   3,   7,   7,   3;
   52 |   4,  13,  18,  13,   4;
  130 |   5,  21,  39,  39,  21,   5;
  319 |   6,  31,  74,  97,  74,  31,   6;
  776 |   7,  43, 127, 211, 211, 127,  43,  7;
  ... |   ...

Mathematica

T[n_, k_] := T[n, k] = Piecewise[{{0, (n == 0 && k == 0) || k == -1 || n - k == -1}, {T[n - 1, k] + T[n - 1, k - 1] + T[n - 2, k - 1] + 1, n > 0 && k >= 0 && n - k >= 0}}];
TableForm[Table[T[n, k], {n, 0, 10}, {k, 0, n}], TableAlignments -> Center, TableSpacing -> {1, 1}]

Crossrefs

Cf. A391272 (row sums).

Sequence in context: A264506 A085204 A228527 * A055375 A091533 A055376

Adjacent sequences: A391315 A391316 A391317 * A391319 A391320 A391321

Keyword

nonn,easy,tabl

Author

Leonidas Liponis, Dec 06 2025

Status

approved