A299646 a(n) = Sum_{k = n..2*n+1} k^2.

1, 14, 54, 135, 271, 476, 764, 1149, 1645, 2266, 3026, 3939, 5019, 6280, 7736, 9401, 11289, 13414, 15790, 18431, 21351, 24564, 28084, 31925, 36101, 40626, 45514, 50779, 56435, 62496, 68976, 75889, 83249, 91070, 99366, 108151, 117439, 127244, 137580, 148461, 159901

Offset: 0

Author

Bruno Berselli, Feb 20 2018

Keywords

• nonn
• easy

Comments

Inverse binomial transform is 1, 13, 27, 14, 0, 0, 0, ... (0 continued).

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

Subsequence of A008854, A047388, A174070 (after 1).

Cf. A050409: Sum_{k = n..2*n} k^2; A050410: Sum_{k = n..2*n-1} k^2.

Programs

GAP

List([0..50], n -> (n+2)*(14*n^2+11*n+3)/6);

Magma

[(n+2)*(14*n^2+11*n+3)/6: n in [0..50]];

Maple

seq((n + 2)*(14*n^2 + 11*n + 3)/6, n=0..50); # Peter Luschny, Feb 21 2018

Mathematica

Table[(n + 2) (14 n^2 + 11 n + 3)/6, {n, 0, 50}]
(* Second program: *)
LinearRecurrence[{4, -6, 4, -1}, {1, 14, 54, 135}, 41] (* Jean-François Alcover, Feb 21 2018 *)

Maxima

makelist((n+2)*(14*n^2+11*n+3)/6, n, 0, 50);

PARI

a(n)=(n+2)*(14*n^2+11*n+3)/6 \\ Charles R Greathouse IV, Feb 21 2018

PARI

Vec((1 + 10*x + 4*x^2 - x^3)/(1 - x)^4 + O(x^60)) \\ Colin Barker, Feb 22 2018

Sage

[(n+2)*(14*n^2+11*n+3)/6 for n in (0..50)]

Formula

O.g.f.: (1 + 10*x + 4*x^2 - x^3)/(1 - x)^4.

E.g.f.: (6 + 78*x + 81*x^2 + 14*x^3)*exp(x)/6.

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

a(n) = (n + 2)*(14*n^2 + 11*n + 3)/6. Therefore:

a(6*k + r) = 504*k^3 + 18*(14*r + 13)*k^2 + (42*r^2 + 78*r + 25)*k + a(r), with 0 <= r <= 5. Example: for r=5, a(6*k + 5) = (6*k + 7)*(84*k^2 + 151*k + 68).