A100147 Structured icosidodecahedral numbers.

1, 30, 135, 364, 765, 1386, 2275, 3480, 5049, 7030, 9471, 12420, 15925, 20034, 24795, 30256, 36465, 43470, 51319, 60060, 69741, 80410, 92115, 104904, 118825, 133926, 150255, 167860, 186789, 207090, 228811, 252000, 276705, 302974, 330855, 360396, 391645, 424650

Offset: 1

Author

James A. Record (james.record(AT)gmail.com), Nov 07 2004

Keywords

• easy
• nonn

Comments

Equals row sums of triangle A143254 & binomial transform of [1, 29, 76, 48, 0, 0, 0, ...]. - Gary W. Adamson, Aug 02 2008

Apart from offset, same as A079588.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..5000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A100146, A100148 for adjacent structured Archimedean solids; and A100145 for more on structured polyhedral numbers.

Cf. also A079588.

Programs

Magma

[(1+(n-1))*(1+2*(n-1))*(1+4*(n-1)): n in [1..40]]; // Vincenzo Librandi, Jul 19 2011

Mathematica

Table[(48*n^3 - 60*n^2 + 18*n)/6, {n,1,50}] (* G. C. Greubel, Oct 18 2018 *)

PARI

a(n)=n*(8*n^2-10*n+3) \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = (1/6)*(48*n^3 - 60*n^2 + 18*n).

a(n) = A079588(n-1) = n*(2*n-1)*(4*n-3). - R. J. Mathar, Sep 02 2008

From Jaume Oliver Lafont, Sep 08 2009: (Start)

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

G.f.: x*(1 + 26*x + 21*x^2)/(1-x)^4. (End)

E.g.f.: x*(1 + 14*x + 8*x^2)*exp(x). - G. C. Greubel, Oct 18 2018

From Amiram Eldar, Sep 20 2022: (Start)

Sum_{n>=1} 1/a(n) = Pi/3.

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