A005904 Centered dodecahedral numbers. (Formerly M5239)

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

0,2

References

Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 137.

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).

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

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 *)

Crossrefs

Cf. A001620, A006566.

Sequence in context: A393662 A283552 A007260 * A207078 A182588 A199900

Adjacent sequences: A005901 A005902 A005903 * A005905 A005906 A005907

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved