A365904 Triangle read by rows T(n,k) = n^2 - binomial(k+1,2), n>=1, k

1, 4, 3, 9, 8, 6, 16, 15, 13, 10, 25, 24, 22, 19, 15, 36, 35, 33, 30, 26, 21, 49, 48, 46, 43, 39, 34, 28, 64, 63, 61, 58, 54, 49, 43, 36, 81, 80, 78, 75, 71, 66, 60, 53, 45, 100, 99, 97, 94, 90, 85, 79, 72, 64, 55, 121, 120, 118, 115, 111, 106, 100, 93, 85, 76, 66

Offset: 1

Author

Joan Llobera Querol, Sep 22 2023

Keywords

• nonn
• tabl

Comments

T(n,k) is the number of points in a rhombus that has 1 point in the first row, then 2 in the second, and following until the n-th row with n points, and then n-1 in the following row, n-2 in the following to end with a row with k+1 points.

T(n,0) are the perfect squares (A000290).

T(n,n-1) are the triangular numbers (A000217).

Examples

			Triangle begins:
    1;
    4,  3;
    9,  8,  6;
   16, 15, 13, 10;
   25, 24, 22, 19, 15;
   36, 35, 33, 30, 26, 21;
   49, 48, 46, 43, 39, 34, 28;
   64, 63, 61, 58, 54, 49, 43, 36;
   81, 80, 78, 75, 71, 66, 60, 53, 45;
  100, 99, 97, 94, 90, 85, 79, 72, 64, 55;
  ...
		

Links

• Joan Llobera Querol, Table of n, a(n) for n = 1..10000
• Zach Wissner-Gross, Can You Shape the Peloton?, Fiddler on the Proof, Sep 22, 2023.

Crossrefs

Row sums give A004068.

Cf. A214859.

Formula

G.f.: x*(1 + x - 4*x^2*y + x^3*y^2 + x^4*y^2)/((1 - x)^3*(1 - x*y)^3). - Stefano Spezia, Oct 05 2023