A232713 Doubly pentagonal numbers: a(n) = n*(3*n-2)*(3*n-1)*(3*n+1)/8.

0, 1, 35, 210, 715, 1820, 3876, 7315, 12650, 20475, 31465, 46376, 66045, 91390, 123410, 163185, 211876, 270725, 341055, 424270, 521855, 635376, 766480, 916895, 1088430, 1282975, 1502501, 1749060, 2024785, 2331890, 2672670, 3049501, 3464840, 3921225, 4421275

Offset: 0

Author

Bruno Berselli, Nov 28 2013

Keywords

• nonn
• easy

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

Cf. A000326, A000332.

Cf. similar sequences: A000583 for A000290(A000290(n)); A002817 for A000217(A000217(n)); A063249 for A000384(A000384(n)).

Cf. A236770, A260810.

Programs

Magma

[n*(3*n-2)*(3*n-1)*(3*n+1)/8: n in [0..40]];

Mathematica

Table[n (3 n - 2) (3 n - 1) (3 n + 1)/8, {n, 0, 40}]

PARI

a(n)=n*(3*n-2)*(3*n-1)*(3*n+1)/8 \\ Charles R Greathouse IV, Oct 07 2015

Formula

G.f.: x*(1 + 30*x + 45*x^2 + 5*x^3) / (1 - x)^5.

a(n) = A000326(A000326(n)) = A000332(3n+1).

From Amiram Eldar, Aug 25 2022: (Start)

Sum_{n>=1} 1/a(n) = 4 + 2*Pi/sqrt(3) - 6*log(3).

Sum_{n>=1} (-1)^(n+1)/a(n) = 32*log(2)/3 - 4*Pi/(3*sqrt(3)) - 4. (End)