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

0,2

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

Links

Table of n, a(n) for n=0..43.

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

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)

Example

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.

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A107614 A317758 A010915 * A260384 A126360 A264545

Adjacent sequences: A352112 A352113 A352114 * A352116 A352117 A352118

Keyword

nonn,easy

Author

David James Sycamore, Mar 05 2022

Status

approved