%I A001082
%S A001082 0,1,5,8,16,21,33,40,56,65,85,96,120,133,161,176,208,225,261,280,320,
%T A001082 341,385,408,456,481,533,560,616,645,705,736,800,833,901,936,1008,1045,
%U A001082 1121,1160,1240,1281,1365,1408,1496,1541,1633,1680,1776,1825,1925,1976
%N A001082 a(n) = n(3n-4)/4 if n even, (n-1)(3n+1)/4 if n odd.
%C A001082 3*a(n)+1 is a perfect square.
%C A001082 Could also be called generalized octagonal numbers, or n(3n-2) for n=0, 
               +- 1, +- 2,.... Cf. A001318, generalized pentagonal numbers. - Matthew 
               Vandermast (ghodges14(AT)comcast.net), Apr 10 2003
%C A001082 n^2 - n - floor(n/2)^2.
%C A001082 Sequence allows us to find X values of the equation: 3*X^3 + X^2 = Y^2. 
               - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Nov 06 2007
%C A001082 Number of units of a(n) belongs to a periodic sequence: 0, 1, 5, 8, 6, 
               1, 3, 0, 6, 5, 5, 6, 0, 3, 1, 6, 8, 5, 1, 0. [From Mohamed Bouhamida 
               (bhmd95(AT)yahoo.fr), Sep 04 2009]
%H A001082 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A001082 R. Stephan, <a href="http://www.ark.in-berlin.de/A001082.ps">On the solutions 
               to 'px+1 is square'</a>
%F A001082 G.f.: sum_{n=0..inf} (-1)^n*[x^(a(2n+1)) + x^(a(2n+2))] = 1/1 - (x-x^2)/
               1 - (x^2-x^4)/1 - (x^3-x^6)/1 -...- (x^k - x^(2k))/1 -... (continued 
               fraction where k=1..inf). - Paul D. Hanna (pauldhanna(AT)juno.com), 
               Aug 16 2002
%F A001082 a(2n)=n(3n+2), a(2n+1)=3*n^2+4n+1. - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), 
               Nov 06 2007
%F A001082 a(n+1) = ceil(n/2)^2+A046092([n/2]).
%F A001082 a(2n)=n(3n-2)=A000567(n), a(2n+1)=n(3n+2)=A045944(n). - Mohamed Bouhamida 
               (bhmd95(AT)yahoo.fr), Nov 06 2007
%F A001082 O.g.f.: -x^2*(x^2+4*x+1)/((x-1)^3*(1+x)^2). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Apr 15 2008
%F A001082 a(n+1)-a(n)=A022998(n). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Apr 15 2008
%o A001082 (PARI) a(n)=if(n%2,(n-1)*(3*n+1)/4,n*(3*n-4)/4)
%Y A001082 Partial sums of A022998.
%Y A001082 Cf. A005563, A046092.
%Y A001082 Sequence in context: A141536 A065905 A126695 this_sequence A030006 A088586 
               A073136
%Y A001082 Adjacent sequences: A001079 A001080 A001081 this_sequence A001083 A001084 
               A001085
%K A001082 nonn,easy
%O A001082 1,3
%A A001082 N. J. A. Sloane (njas(AT)research.att.com) and Tom Duff
%E A001082 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Sep 19 2000
%E A001082 More terms from Ralf Stephan (ralf(AT)ark.in-berlin.de), Jul 25 2003
%E A001082 Some of the formulae were corrected by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Apr 15 2008