A143183 Triangle T(n,k) = 1 + (2+n)*abs(n-2*k), read by rows, for 0 <= k <= n.

1, 4, 4, 9, 1, 9, 16, 6, 6, 16, 25, 13, 1, 13, 25, 36, 22, 8, 8, 22, 36, 49, 33, 17, 1, 17, 33, 49, 64, 46, 28, 10, 10, 28, 46, 64, 81, 61, 41, 21, 1, 21, 41, 61, 81, 100, 78, 56, 34, 12, 12, 34, 56, 78, 100, 121, 97, 73, 49, 25, 1, 25, 49, 73, 97, 121

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Oct 17 2008

Keywords

• nonn
• tabl
• easy

Examples

			Triangle begins as:
    1;
    4,  4;
    9,  1,  9;
   16,  6,  6, 16;
   25, 13,  1, 13, 25;
   36, 22,  8,  8, 22, 36;
   49, 33, 17,  1, 17, 33, 49;
   64, 46, 28, 10, 10, 28, 46, 64;
   81, 61, 41, 21,  1, 21, 41, 61, 81;
  100, 78, 56, 34, 12, 12, 34, 56, 78, 100;
  121, 97, 73, 49, 25,  1, 25, 49, 73,  97, 121;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A000290, A005843, A007590, A143182.

Programs

Magma

[1+(n+2)*Abs(n-2*k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 23 2024

Maple

A143183 := proc(n,k)
        1+(2+n)*abs(n-2*m) ;
end proc: # R. J. Mathar, Jul 12 2012

Mathematica

T[n_, m_]:= 1 + Abs[(n-m+1)^2 - (m+1)^2];
Table[T[n, m], {n,0,12}, {m,0,n}]//Flatten

SageMath

flatten([[1+(n+2)*abs(n-2*k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Apr 23 2024

Formula

T(n, k) = 1 + (2+n)*abs(n-2*k), for 0 <= k <= n.

T(n, k) = T(n, n-k).

Sum_{k=0..n} T(n, k) = (n+2)*A007590(n+1) + n + 1 (row sums).

From G. C. Greubel, Apr 23 2024: (Start)

T(n, 0) = A000290(n+1).

T(2*n-1, n) = A005843(n+1), n >= 1.

Sum_{k=0..n} (-1)^k*T(n, k) = (1/2)*(1 + (-1)^n)*((n^2 + 3*n + 3) - (-1)^(n/2)*(n + 2)). (End)

Extensions

Row sums corrected by R. J. Mathar, Jul 12 2012