A014633 - OEIS

%I #48 Jan 13 2024 03:33:27

%S 0,12,22,70,92,176,210,330,376,532,590,782,852,1080,1162,1426,1520,

%T 1820,1926,2262,2380,2752,2882,3290,3432,3876,4030,4510,4676,5192,

%U 5370,5922,6112,6700,6902,7526,7740,8400,8626,9322

%N Even pentagonal numbers.

%H Vincenzo Librandi, <a href="/A014633/b014633.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).

%F 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

%F From _Ant King_, Aug 16 2011: (Start)

%F a(n) = a(n-1) +2*a(n-2) -2*a(n-3) -a(n-4) +a(n-5).

%F a(n) = 48+2*a(n-2)-a(n-4).

%F a(n) = 1/8*(1-3*(-1)^(n+1)+12*(n+1))*(1-(-1)^(n+1)+4*(n+1)).(End)

%F 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

%t LinearRecurrence[{1,2,-2,-1,1},{0,12,22,70,92},40] (* _Harvey P. Dale_, Aug 26 2014 *)

%t Select[PolygonalNumber[5,Range[0,100]],EvenQ] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Jan 15 2017 *)

%o (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

%o (PARI) lista(nn) = {forstep (n=0, nn, 2, if (ispolygonal(n, 5), print1(n, ", ")););} \\ _Michel Marcus_, Jun 20 2015

%Y Cf. A000326, A014632.

%K nonn,easy

%O 0,2

%A _Mohammad K. Azarian_

%E More terms from _Patrick De Geest_