A368343 Square array T(n,k), n >= 3, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} k^(n-j) * floor(j/3).

1, 1, 1, 1, 2, 1, 1, 3, 3, 2, 1, 4, 7, 5, 2, 1, 5, 13, 16, 7, 2, 1, 6, 21, 41, 34, 9, 3, 1, 7, 31, 86, 125, 70, 12, 3, 1, 8, 43, 157, 346, 377, 143, 15, 3, 1, 9, 57, 260, 787, 1386, 1134, 289, 18, 4, 1, 10, 73, 401, 1562, 3937, 5547, 3405, 581, 22, 4

Offset

3,5

Links

Seiichi Manyama, Antidiagonals n = 3..142, flattened

Formula

T(n,k) = T(n-3,k) + Sum_{j=0..n-3} k^j.

T(n,k) = 1/(k-1) * Sum_{j=0..n} floor(k^j/(k^2+k+1)) = Sum_{j=0..n} floor(k^j/(k^3-1)) for k > 1.

T(n,k) = (k+1)*T(n-1,k) - k*T(n-2,k) + T(n-3,k) - (k+1)*T(n-4,k) + k*T(n-5,k).

G.f. of column k: x^3/((1-x) * (1-k*x) * (1-x^3)).

T(n,k) = 1/(k-1) * (floor(k^(n+1)/(k^3-1)) - floor((n+1)/3)) for k > 1.

Example

Square array begins:
  1,  1,   1,    1,    1,     1,     1, ...
  1,  2,   3,    4,    5,     6,     7, ...
  1,  3,   7,   13,   21,    31,    43, ...
  2,  5,  16,   41,   86,   157,   260, ...
  2,  7,  34,  125,  346,   787,  1562, ...
  2,  9,  70,  377, 1386,  3937,  9374, ...
  3, 12, 143, 1134, 5547, 19688, 56247, ...

Prog

(PARI) T(n, k) = sum(j=0, n, k^(n-j)*(j\3));

Crossrefs

Columns k=0..4 give A002264, A130518, A178455, A368344, A368345.

Cf. A055129, A368296.

Sequence in context: A099509 A153859 A363394 * A131336 A052253 A271453

Adjacent sequences: A368340 A368341 A368342 * A368344 A368345 A368346

Keyword

nonn,tabl

Author

Seiichi Manyama, Dec 22 2023

Status

approved