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A104585 a(n) = (1/2) * ( 3*n^2 - n*(-1)^n ).

%I A104585 #51 Feb 22 2022 03:36:10
%S A104585 0,2,5,15,22,40,51,77,92,126,145,187,210,260,287,345,376,442,477,551,
%T A104585 590,672,715,805,852,950,1001,1107,1162,1276,1335,1457,1520,1650,1717,
%U A104585 1855,1926,2072,2147,2301,2380,2542,2625,2795,2882,3060,3151,3337,3432,3626
%N A104585 a(n) = (1/2) * ( 3*n^2 - n*(-1)^n ).
%C A104585 Previous name was: Pentagonal wave sequence of the second kind.
%C A104585 Even-indexed terms are pentagonal numbers with even index in A000326. Odd-indexed terms are second pentagonal numbers with odd index in A005449.
%C A104585 A104584, pentagonal wave sequence of the first kind; switches odd and even applications and vice versa in A104585. The pentagonal wave triangle, A104586, has A104584 in odd columns and A104585 in even columns.
%C A104585 Integer values of (n+1)(2n+1)/3 in order of appearance. - _Wesley Ivan Hurt_, Sep 17 2013
%C A104585 Exponents of q in the identity 1 - Sum_{n >= 0} ( q^(3*n+2)*Product_{k = 1..n} (1 - q^(4*k-1)) ) = 1 - q^2 - q^5 + q^15 + q^22 - q^40 - q^51 + + - - .... Compare with Euler's pentagonal number theorem: Product_{n >= 1} (1 - q^n) = 1 - Sum_{n >= 1} ( q^n*Product_{k = 1..n-1} (1 - q^k) ) = 1 - q - q^2 + q^5 + q^7 - q^12 - q^15  + + - - .... - _Peter Bala_, Dec 03 2020
%H A104585 Vincenzo Librandi, <a href="/A104585/b104585.txt">Table of n, a(n) for n = 0..1000</a>
%H A104585 Wikipedia, <a href="https://en.wikipedia.org/wiki/Pentagonal_number_theorem">Pentagonal number theorem</a>.
%H A104585 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A104585 a(n) = (1/2) * ( 3*n^2 - n*(-1)^n ). - _Ralf Stephan_, Nov 13 2010
%F A104585 G.f.: x*(2+3*x+6*x^2+x^3)/(1-x)^3/(1+x)^2. - _Colin Barker_, Feb 13 2012
%F A104585 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). - _Vincenzo Librandi_, Apr 04 2013
%F A104585 From _Amiram Eldar_, Feb 22 2022: (Start)
%F A104585 Sum_{n>=1} 1/a(n) = 4*log(2) - Pi/sqrt(3).
%F A104585 Sum_{n>=1} (-1)^(n+1)/a(n) = 2*Pi/sqrt(3) - 3*log(3). (End)
%e A104585 a(5) = 40 = A005449(5), a second pentagonal number.
%e A104585 a(6) = 51 = A000326(6), a pentagonal number.
%t A104585 Table[(1/2) (3 n^2 - n (-1)^n), {n, 0, 100}] (* _Vincenzo Librandi_, Apr 04 2013 *)
%t A104585 LinearRecurrence[{1,2,-2,-1,1},{0,2,5,15,22},50] (* _Harvey P. Dale_, Sep 14 2015 *)
%o A104585 (Magma) I:=[0, 2, 5, 15, 22]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..60]]; // _Vincenzo Librandi_, Apr 04 2013
%Y A104585 Cf. A000326, A005449, A049452, A033570, A049453, A033568, A104586.
%K A104585 nonn
%O A104585 0,2
%A A104585 _Gary W. Adamson_, Mar 17 2005
%E A104585 More terms from _Colin Barker_, Feb 13 2012
%E A104585 Better name, using formula from _Ralf Stephan_, _Joerg Arndt_, Sep 17 2013