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

Author

Gary W. Adamson, Sep 19 2008

Keywords

• nonn
• tabl

Comments

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

Examples

			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;
  ...
		

Links

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

Crossrefs

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

Programs

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 *)

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

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)