A141433 Triangle T(n, k) = (k-1)*(3*n-k), read by rows.

0, 0, 4, 0, 7, 12, 0, 10, 18, 24, 0, 13, 24, 33, 40, 0, 16, 30, 42, 52, 60, 0, 19, 36, 51, 64, 75, 84, 0, 22, 42, 60, 76, 90, 102, 112, 0, 25, 48, 69, 88, 105, 120, 133, 144, 0, 28, 54, 78, 100, 120, 138, 154, 168, 180

Offset: 1

Author

Roger L. Bagula and Gary W. Adamson, Aug 06 2008

Keywords

• nonn
• easy
• tabl

Examples

			Triangle begins as:
  0;
  0,  4;
  0,  7, 12;
  0, 10, 18, 24;
  0, 13, 24, 33,  40;
  0, 16, 30, 42,  52,  60;
  0, 19, 36, 51,  64,  75,  84;
  0, 22, 42, 60,  76,  90, 102, 112;
  0, 25, 48, 69,  88, 105, 120, 133, 144;
  0, 28, 54, 78, 100, 120, 138, 154, 168, 180;
		

Links

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

Crossrefs

Cf. A304993 (row sums).

Programs

Magma

[(k-1)*(3*n-k): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 01 2021

Maple

A141433 := proc(n,m) (m-1)*(3*n-m) ; end proc:
seq(seq(A141433(n,m),m=1..n),n=1..18) ; # R. J. Mathar, Sep 14 2011

Mathematica

Flatten[Table[(m-1)(3n-m),{n,10},{m,n}]] (* Harvey P. Dale, Feb 04 2016 *)

Sage

flatten([[(k-1)*(3*n-k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Apr 01 2021

Formula

G.f.: Sum_{k>=0} Sum_{n>=0} T(n,k)*x^n*y^k = y^2*x*(2*x+1-3*y)/((1-y)^3*(x-1)^2). (G.f. for the full array, not just the triangular subspace) - R. J. Mathar, Feb 19 2020

Sum_{k=1..n} T(n, k) = A304993(n-1) = (n-1)*n*(7*n -2)/6. - G. C. Greubel, Apr 01 2021