A144680 Triangle read by rows, lower half of an array formed by A004736 * A144328 (transform).

1, 2, 3, 3, 5, 7, 4, 7, 11, 14, 5, 9, 15, 21, 25, 6, 11, 19, 28, 36, 41, 7, 13, 23, 35, 47, 57, 63, 8, 15, 27, 42, 58, 73, 85, 92, 9, 17, 31, 49, 69, 89, 107, 121, 129, 10, 19, 35, 56, 80, 105, 129, 150, 166, 175

Offset

1,2

Comments

Triangle read by rows, lower half of an array formed by A004736 * A144328 (transform).

Links

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Formula

Sum_{k=1..n} T(n, k) = A006008(n).

From G. C. Greubel, Oct 18 2021: (Start)

T(n, k) = (1/6)*( 3*(k^2 - k + 2)*n - k*(k-1)*(2*k-1) ).

T(n, n) = A004006(n).

T(n, n-1) = A050407(n+2).

T(n, n-2) = A027965(n-1) = A074742(n-2). (End)

Example

The array is formed by A004736 * A144328 (transform) where A004736 = the natural number decrescendo triangle and A144328 = a crescendo triangle. First few rows of the array =
  1, 1,  1,  1,  1,  1, ...
  2, 3,  3,  3,  3,  3, ...
  3, 5,  7,  7,  7,  7, ...
  4, 7, 11, 14, 14, 14, ...
  5, 9, 15, 21, 25, 25, ...
  ...
Triangle begins as:
   1;
   2,  3;
   3,  5,  7;
   4,  7, 11, 14;
   5,  9, 15, 21, 25;
   6, 11, 19, 28, 36,  41;
   7, 13, 23, 35, 47,  57,  63;
   8, 15, 27, 42, 58,  73,  85,  92;
   9, 17, 31, 49, 69,  89, 107, 121, 129;
  10, 19, 35, 56, 80, 105, 129, 150, 166, 175;
  ...

Mathematica

T[n_, k_]:= (3*(k^2-k+2)*n - k*(k-1)*(2*k-1))/6;
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Oct 18 2021 *)

Prog

(Sage)
def A144680(n, k): return (3*(k^2-k+2)*n - k*(k-1)*(2*k-1))/6
flatten([[A144680(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Oct 18 2021

Crossrefs

Cf. A004006, A006008, A027965, A050407, A074742, A144328.

Sequence in context: A092557 A333295 A081493 * A209702 A077558 A122444

Adjacent sequences: A144677 A144678 A144679 * A144681 A144682 A144683

Keyword

nonn,tabl

Author

Gary W. Adamson, Sep 19 2008

Status

approved