A005904 Centered dodecahedral numbers.

1, 33, 155, 427, 909, 1661, 2743, 4215, 6137, 8569, 11571, 15203, 19525, 24597, 30479, 37231, 44913, 53585, 63307, 74139, 86141, 99373, 113895, 129767, 147049, 165801, 186083, 207955, 231477, 256709, 283711, 312543, 343265, 375937, 410619, 447371, 486253, 527325

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Amiram Eldar, Table of n, a(n) for n = 0..10000
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
• Boon K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558; alternative link.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A001620, A006566.

Programs

Maple

A005904:=(z+1)*(z**2+28*z+1)/(z-1)**4; [Conjectured by Simon Plouffe in his 1992 dissertation.]

Mathematica

a[n_] := (2*n + 1) * (5*n^2 + 5*n + 1); Array[a, 30, 0] (* Amiram Eldar, Sep 12 2022 *)

Formula

a(n) = (2*n+1)*(5*n^2+5*n+1).

Sum_{n>=0} 1/a(n) = -psi((5+sqrt(5))/10) - psi((5-sqrt(5))/10) - 2*gamma - 4*log(2), where psi is the digamma function and gamma is Euler's constant (A001620). - Amiram Eldar, Sep 12 2022

E.g.f.: exp(x)*(1 + 32*x + 45*x^2 + 10*x^3). - Stefano Spezia, Jun 06 2025