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A100147 Structured icosidodecahedral numbers.

%I A100147 #30 Sep 20 2022 02:37:43
%S A100147 1,30,135,364,765,1386,2275,3480,5049,7030,9471,12420,15925,20034,
%T A100147 24795,30256,36465,43470,51319,60060,69741,80410,92115,104904,118825,
%U A100147 133926,150255,167860,186789,207090,228811,252000,276705,302974,330855,360396,391645,424650
%N A100147 Structured icosidodecahedral numbers.
%C A100147 Equals row sums of triangle A143254 & binomial transform of [1, 29, 76, 48, 0, 0, 0, ...]. - _Gary W. Adamson_, Aug 02 2008
%C A100147 Apart from offset, same as A079588.
%H A100147 Vincenzo Librandi, <a href="/A100147/b100147.txt">Table of n, a(n) for n = 1..5000</a>
%H A100147 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A100147 a(n) = (1/6)*(48*n^3 - 60*n^2 + 18*n).
%F A100147 a(n) = A079588(n-1) = n*(2*n-1)*(4*n-3). - _R. J. Mathar_, Sep 02 2008
%F A100147 From _Jaume Oliver Lafont_, Sep 08 2009: (Start)
%F A100147 a(n) = (1+(n-1))*(1+2*(n-1))*(1+4*(n-1)).
%F A100147 G.f.: x*(1 + 26*x + 21*x^2)/(1-x)^4. (End)
%F A100147 E.g.f.: x*(1 + 14*x + 8*x^2)*exp(x). - _G. C. Greubel_, Oct 18 2018
%F A100147 From _Amiram Eldar_, Sep 20 2022: (Start)
%F A100147 Sum_{n>=1} 1/a(n) = Pi/3.
%F A100147 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)
%t A100147 Table[(48*n^3 - 60*n^2 + 18*n)/6, {n,1,50}] (* _G. C. Greubel_, Oct 18 2018 *)
%o A100147 (Magma) [(1+(n-1))*(1+2*(n-1))*(1+4*(n-1)): n in [1..40]]; // _Vincenzo Librandi_, Jul 19 2011
%o A100147 (PARI) a(n)=n*(8*n^2-10*n+3) \\ _Charles R Greathouse IV_, Oct 07 2015
%Y A100147 Cf. A100146, A100148 for adjacent structured Archimedean solids; and A100145 for more on structured polyhedral numbers.
%Y A100147 Cf. also A079588.
%K A100147 easy,nonn
%O A100147 1,2
%A A100147 James A. Record (james.record(AT)gmail.com), Nov 07 2004