A014633 Even pentagonal numbers.
0, 12, 22, 70, 92, 176, 210, 330, 376, 532, 590, 782, 852, 1080, 1162, 1426, 1520, 1820, 1926, 2262, 2380, 2752, 2882, 3290, 3432, 3876, 4030, 4510, 4676, 5192, 5370, 5922, 6112, 6700, 6902, 7526, 7740, 8400, 8626, 9322
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..10000
- Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).
Programs
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Magma
[1/8*(1-3*(-1)^(n+1)+12*(n+1))*(1-(-1)^(n+1)+4*(n+1)): n in [0..40]]; // Vincenzo Librandi, Aug 17 2011
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Mathematica
LinearRecurrence[{1,2,-2,-1,1},{0,12,22,70,92},40] (* Harvey P. Dale, Aug 26 2014 *) Select[PolygonalNumber[5,Range[0,100]],EvenQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 15 2017 *) -
PARI
lista(nn) = {forstep (n=0, nn, 2, if (ispolygonal(n, 5), print1(n, ", ")););} \\ Michel Marcus, Jun 20 2015
Formula
G.f.: 2*(6+5*x+12*x^2+x^3)/((1+x)^2*(1-x)^3). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009, corrected by R. J. Mathar, Sep 16 2009
From Ant King, Aug 16 2011: (Start)
a(n) = a(n-1) +2*a(n-2) -2*a(n-3) -a(n-4) +a(n-5).
a(n) = 48+2*a(n-2)-a(n-4).
a(n) = 1/8*(1-3*(-1)^(n+1)+12*(n+1))*(1-(-1)^(n+1)+4*(n+1)).(End)
Sum_{n>=1} 1/a(n) = 3*log(3)/2 - (1/sqrt(3)+1/4)*Pi - sqrt(3)*log(2-sqrt(3))/2. - Amiram Eldar, Jan 13 2024
Extensions
More terms from Patrick De Geest