A014370 If n = binomial(b,2) + binomial(c,1), b > c >= 0 then a(n) = binomial(b+1,3) + binomial(c+1,2).

1, 2, 4, 5, 7, 10, 11, 13, 16, 20, 21, 23, 26, 30, 35, 36, 38, 41, 45, 50, 56, 57, 59, 62, 66, 71, 77, 84, 85, 87, 90, 94, 99, 105, 112, 120, 121, 123, 126, 130, 135, 141, 148, 156, 165, 166, 168, 171, 175, 180, 186, 193, 201, 210, 220, 221, 223, 226, 230, 235, 241, 248, 256, 265, 275, 286

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• tabl

Examples

			The triangle starts:
  1
  2 4
  5 7 10
  11 13 16 20
  21 23 26 30 35
		

References

• W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge, 1993, p. 159.

Links

• Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of triangle, flattened).

Crossrefs

Cf. A002260, A000292 (main diagonal), A000217, A014368, A014369, A006046, A050407 (1st column), A005581 (subdiagonal), A071239 (row sums), A212013.

Programs

Maple

a := 0: for i from 1 to 15 do for j from 1 to i do a := a+j: printf(`%d,`,a); od:od:

Mathematica

A014370[n_, k_] := Binomial[n + 1, 3] + Binomial[k + 1, 2];
Table[A014370[n, k], {n, 12}, {k, n}] (* Paolo Xausa, Mar 11 2025 *)

Formula

a(n) = Sum_{m = 1..n} b(m), b(m) = 1,1,2,1,2,3,1,2,3,4,... = A002260. - Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)

a(n*(n+1)/2+m) = n*(n+1)*(n+2)/6 + m*(m+1)/2 = A000292(n)+ A000217(m), m = 0...n+1, n = 1, 2, 3.. - Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)

a(n) = a(n-1) + A002260(n). As a triangle with n >= k >= 1: a(n, k) = a(n-1, k) + (n-1)*n/2 = a(n, k-1) + k = (n^3-n+3k^2+3k)/6. - Henry Bottomley, Nov 14 2001

a(n) = b(n) * (b(n) + 1) * (b(n) + 2) / 6 + c(n) * (c(n) + 1) / 2, where b(n) = [sqrt(2 * n) - 1/2] and c(n) = n - b(n) * (b(n) + 1) / 2. - Robert A. Stump (bee_ess107(AT)msn.com), Sep 20 2002

As a triangle, T(n,k) = binomial(n+1, 3) + binomial(k+1,2). - Franklin T. Adams-Watters, Jan 27 2014

Extensions

More terms from James Sellers, Feb 05 2000