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

1,2

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

Links

Table of n, a(n) for n=1..55.

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

Example

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;
  ...

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

Crossrefs

Cf. A001296 (row sums).

Sequence in context: A028987 A284615 A186709 * A225059 A190899 A173671

Adjacent sequences: A143034 A143035 A143036 * A143038 A143039 A143040

Keyword

nonn,tabl,easy

Author

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

Status

approved