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

0,3

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).

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)

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 *)

Prog

(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)]
(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

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).

Sequence in context: A299580 A289350 A100930 * A274610 A290800 A251930

Adjacent sequences: A260807 A260808 A260809 * A260811 A260812 A260813

Keyword

nonn,easy

Author

Bruno Berselli, Jul 31 2015

Status

approved