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

1,2

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).

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)

Mathematica

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

Prog

(Magma) [(1+(n-1))*(1+2*(n-1))*(1+4*(n-1)): n in [1..40]]; // Vincenzo Librandi, Jul 19 2011
(PARI) a(n)=n*(8*n^2-10*n+3) \\ Charles R Greathouse IV, Oct 07 2015

Crossrefs

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

Cf. also A079588.

Sequence in context: A221522 A291582 A079588 * A117750 A348828 A158462

Adjacent sequences: A100144 A100145 A100146 * A100148 A100149 A100150

Keyword

easy,nonn

Author

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

Status

approved