A304993 a(n) = n*(n + 1)*(7*n + 5)/6.

0, 4, 19, 52, 110, 200, 329, 504, 732, 1020, 1375, 1804, 2314, 2912, 3605, 4400, 5304, 6324, 7467, 8740, 10150, 11704, 13409, 15272, 17300, 19500, 21879, 24444, 27202, 30160, 33325, 36704, 40304, 44132, 48195, 52500, 57054, 61864, 66937, 72280, 77900, 83804, 89999, 96492

Offset: 0

Author

Bruno Berselli, May 23 2018

Keywords

• nonn
• easy

Comments

The sequence provides the sums of the triangular numbers from A000217(n) to A000217(2*n).

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A000217, A255211.

Partial sums of A022265.

Cf. A045943: Sum_{k = n..2*n} k.

Cf. A050409: Sum_{k = n..2*n} k^2.

Row sums of the triangle in A141433.

Programs

Mathematica

Table[n (n + 1) (7 n + 5)/6, {n, 0, 50}]
LinearRecurrence[{4,-6,4,-1},{0,4,19,52},50] (* Harvey P. Dale, May 03 2023 *)

PARI

concat(0, Vec(x*(4 + 3*x)/(1 - x)^4 + O(x^40))) \\ Colin Barker, May 25 2018

Formula

O.g.f.: x*(4 + 3*x)/(1 - x)^4.

E.g.f.: x*(24 + 33*x + 7*x^2)*exp(x)/6.

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

a(n) = -A255211(-n-1).

a(n) + a(-n) = A016742(n).

a(n) = Sum_{k = n..2*n} k*(k+1)/2.