A352115 Partial sums of the even triangular numbers (A014494).

0, 6, 16, 44, 80, 146, 224, 344, 480, 670, 880, 1156, 1456, 1834, 2240, 2736, 3264, 3894, 4560, 5340, 6160, 7106, 8096, 9224, 10400, 11726, 13104, 14644, 16240, 18010, 19840, 21856, 23936, 26214, 28560, 31116, 33744, 36594, 39520, 42680, 45920, 49406, 52976, 56804

Offset: 0

Author

David James Sycamore, Mar 05 2022

Keywords

• nonn
• easy

Comments

The absolute difference between the n-th partial sum of the odd triangular numbers and the (n-1)-th partial sum of the even triangular numbers is equal to n; see formula.

Partial sums of the even generalized hexagonal numbers. - Omar E. Pol, Mar 05 2022

Examples

			a(0) = 0 because 0 is the first even term in A000217.
a(1) = 6, the sum of the first two even terms in A000217, and so on.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Crossrefs

Cf. A001477, A000217, A000292, A014493, A014494, A352116, A000447.

Programs

Mathematica

LinearRecurrence[{2, 1, -4, 1, 2, -1}, {0, 6, 16, 44, 80, 146}, 50] (* Amiram Eldar, Mar 05 2022 *)

PARI

te(n) = (2*n+1)*(2*n+1-(-1)^n)/2; \\ A014494
a(n) = sum(k=0, n, te(k)); \\ Michel Marcus, Mar 06 2022

Python

def A352115(n): return (n + 1)*(2*n*(n+2) + 3*(n%2))//3 # Chai Wah Wu, Mar 11 2022

Formula

a(n) = Sum_{k=0..n-1} A014494(k) = Sum_{k=0..n-1} (2*k+1)(2*k+1-(-1)^k)/2.

|A352116(n) - a(n-1)| = n.

A352116(n) + a(n-1) = A000447(n), (n >= 1).

From Stefano Spezia, Mar 05 2022: (Start)

a(n) = (n + 1)*(4*n^2 + 8*n + 3 - 3*(-1)^n)/6.

G.f.: 2*x*(3 + 2*x + 3*x^2)/((1 - x)^4*(1 + x)^2). (End)