A260810 a(n) = n^2*(3*n^2 - 1)/2.

0, 1, 22, 117, 376, 925, 1926, 3577, 6112, 9801, 14950, 21901, 31032, 42757, 57526, 75825, 98176, 125137, 157302, 195301, 239800, 291501, 351142, 419497, 497376, 585625, 685126, 796797, 921592, 1060501, 1214550, 1384801, 1572352, 1778337, 2003926, 2250325, 2518776

Offset: 0

Author

Bruno Berselli, Jul 31 2015

Keywords

• nonn
• easy

Comments

Pentagonal numbers with square indices.

After 0, a(k) is a square if k is in A072256.

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Subsequence of A001318 and A245288 (see Formula field).

Cf. A000326, A193218 (first differences).

Cf. A000583 (squares with square indices), A002593 (hexagonal numbers with square indices).

Cf. A232713 (pentagonal numbers with pentagonal indices), A236770 (pentagonal numbers with triangular indices).

Programs

Magma

[n^2*(3*n^2-1)/2: n in [0..40]];

Magma

I:=[0,1,22,117,376]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..40]]; // Vincenzo Librandi, Aug 23 2015

Maple

A260810:=n->n^2*(3*n^2 - 1)/2: seq(A260810(n), n=0..50); # Wesley Ivan Hurt, Apr 25 2017

Mathematica

Table[n^2 (3 n^2 - 1)/2, {n, 0, 40}]
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 1, 22, 117, 376}, 40] (* Vincenzo Librandi, Aug 23 2015 *)

PARI

vector(40, n, n--; n^2*(3*n^2-1)/2)

Sage

[n^2*(3*n^2-1)/2 for n in (0..40)]

Formula

G.f.: x*(1 + x)*(1 + 16*x + x^2)/(1 - x)^5.

a(n) = a(-n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).

a(n) = A245288(2*n^2).

a(n) = A001318(2*n^2-1) with A001318(-1) = 0.

From Amiram Eldar, Aug 25 2022: (Start)

Sum_{n>=1} 1/a(n) = 3 - Pi^2/3 - sqrt(3)*Pi*cot(Pi/sqrt(3)).

Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(3)*Pi*cosec(Pi/sqrt(3)) - Pi^2/6 - 3. (End)