A316466 a(n) = 2*n*(7*n - 3).

0, 8, 44, 108, 200, 320, 468, 644, 848, 1080, 1340, 1628, 1944, 2288, 2660, 3060, 3488, 3944, 4428, 4940, 5480, 6048, 6644, 7268, 7920, 8600, 9308, 10044, 10808, 11600, 12420, 13268, 14144, 15048, 15980, 16940, 17928, 18944, 19988, 21060, 22160, 23288, 24444, 25628, 26840

Offset: 0

Author

Bruno Berselli, Jul 04 2018

Keywords

• nonn
• easy

Comments

This is the case k = 9 of Sum_{i = 2..k} P(i,n) = (k - 1)*n*((k - 2)*n - (k - 6))/4, where P(k,n) = n*((k - 2)*n - (k - 4))/2 (see Crossrefs for similar sequences and "Square array in A139600" in Links section).

14*x + 9 is a square for x = a(n) or x = a(-n).

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Bruno Berselli, Square array in A139600.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A139600, A218471.

Similar sequences (see the first comment): A000096 (k = 3), A045943 (k = 4), A049451 (k = 5), A033429 (k = 6), A167469 (k = 7), A152744 (k = 8), this sequence (k = 9), A152994 (k = 10).

Programs

GAP

List([0..50], n -> 2*n*(7*n-3));

Julia

[2*n*(7*n-3) for n in 0:50] |> println

Magma

[2*n*(7*n-3): n in [0..50]];

Mathematica

Table[2 n (7 n - 3), {n, 0, 50}]
LinearRecurrence[{3,-3,1},{0,8,44},50] (* Harvey P. Dale, Jan 24 2021 *)

Maxima

makelist(2*n*(7*n-3), n, 0, 50);

PARI

vector(50, n, n--; 2*n*(7*n-3))

PARI

concat(0, Vec(4*x*(2 + 5*x)/(1 - x)^3 + O(x^40))) \\ Colin Barker, Jul 05 2018

Python

[2*n*(7*n-3) for n in range(50)]

Sage

[2*n*(7*n-3) for n in (0..50)]

Formula

O.g.f.: 4*x*(2 + 5*x)/(1 - x)^3.

E.g.f.: 2*x*(4 + 7*x)*exp(x).

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

a(n) = 4*A218471(n).