A143037 Triangle read by rows, A000012 * A127773 * A000012. A000012 is an infinite lower triangular matrix with all 1's, A127773 = (1; 0,3; 0,0,6; 0,0,0,10; ...).

1, 3, 4, 6, 9, 10, 10, 16, 19, 20, 15, 25, 31, 34, 35, 21, 36, 46, 52, 55, 56, 28, 49, 64, 74, 80, 83, 84, 36, 64, 85, 100, 110, 116, 119, 120, 45, 81, 109, 130, 145, 155, 161, 164, 165, 55, 100, 136, 164, 185, 200, 210, 216, 219, 220

Offset: 1

Author

Gary W. Adamson & Roger L. Bagula, Jul 18 2008

Keywords

• nonn
• tabl
• easy

Comments

Right border = tetrahedral numbers, left border = triangular numbers.

Alternatively this is the square array A(n,k)

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...

4, 9, 16, 25, 36, 49, 64, 81, 100, 121, ...

10, 19, 31, 46, 64, 85, 109, 136, 166, 199, ...

20, 34, 52, 74, 100, 130, 164, 202, 244, 290, ...

35, 55, 80, 110, 145, 185, 230, 280, 335, 395, ...

56, 83, 116, 155, 200, 251, 308, 371, 440, 515, ...

...

read by antidiagonals where A(n,k) = n*(n^2 + 3*k*n + 3*k^2 - 1)/6 is the sum of n triangular numbers starting at A000217(k). - R. J. Mathar, May 06 2015

Examples

			First few rows of the triangle:
   1;
   3,  4;
   6,  9, 10;
  10, 16, 19,  20;
  15, 25, 31,  34,  35;
  21, 36, 46,  52,  55,  56;
  28, 49, 64,  74,  80,  83,  84;
  36, 64, 85, 100, 110, 116, 119, 120;
  ...
		

Crossrefs

Cf. A001296 (row sums).

Programs

Maple

A143037 := proc(n,k)
    k*(k^2-3*k*n-3*k+3*n^2+6*n+2) / 6 ;
end proc:
seq(seq(A143037(n,k),k=1..n),n=1..12) ; # R. J. Mathar, Aug 31 2022

Mathematica

T[n_,k_] = k*(k^2-3*k*n-3*k+3*n^2+6*n+2) / 6;Table[T[n,k],{n,10},{k,n}]//Flatten (* James C. McMahon, Aug 13 2025 *)

Formula

T(n,k) = k*(k^2-3*k*n-3*k+3*n^2+6*n+2) / 6. - R. J. Mathar, Aug 31 2022