A141431 Triangle T(n,k) = (k-1)*(3*n-k+1), read by rows.

0, 0, 5, 0, 8, 14, 0, 11, 20, 27, 0, 14, 26, 36, 44, 0, 17, 32, 45, 56, 65, 0, 20, 38, 54, 68, 80, 90, 0, 23, 44, 63, 80, 95, 108, 119, 0, 26, 50, 72, 92, 110, 126, 140, 152, 0, 29, 56, 81, 104, 125, 144, 161, 176, 189, 0, 32, 62, 90, 116, 140, 162, 182, 200, 216, 230, 0, 35, 68, 99, 128, 155, 180, 203, 224, 243, 260, 275

Offset: 1

Author

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

Keywords

• nonn
• easy
• tabl

Examples

			Triangle begins as:
  0;
  0,  5;
  0,  8, 14;
  0, 11, 20, 27;
  0, 14, 26, 36,  44;
  0, 17, 32, 45,  56,  65;
  0, 20, 38, 54,  68,  80,  90;
  0, 23, 44, 63,  80,  95, 108, 119;
  0, 26, 50, 72,  92, 110, 126, 140, 152;
  0, 29, 56, 81, 104, 125, 144, 161, 176, 189;
		

Links

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

Crossrefs

Columns: A016789 (k=2), A016933 (k=3), A008591 (k=4).

Cf. A245301 (row sums).

Programs

Magma

[(k-1)*(3*n-k+1): k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 31 2021

Maple

A141431 := proc(n,k)
        (k-1)*(3*n-k+1) ;
end proc:
seq(seq(A141431(n,k),k=1..n),n=1..14) ; # R. J. Mathar, Nov 10 2011

Mathematica

Table[(k-1)*(3*n-k+1), {n,15}, {k,n}]//Flatten (* G. C. Greubel, Mar 31 2021 *)

Sage

flatten([[(k-1)*(3*n-k+1) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 31 2021

Formula

G.f.: Sum_{n>=0} Sum_{k>=0} T(n,k)*x^n*y^k = y^2*x*(x*y-4*y+x+2)/((1-y)^3*(1-x)^2). - R. J. Mathar, Nov 27 2015. x and y swapped to align with standard, 19 Feb 2020

Sum_{k=1..n} T(n, k) = (n-1)*n*(7*n+1)/6 = A245301(n-1). - G. C. Greubel, Mar 31 2021

Extensions

More terms added by G. C. Greubel, Mar 31 2021