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 A001082 Generalized octagonal numbers: n*(3*n-2), n=0, +- 1, +- 2, +-3.... 93

%I

%S 0,1,5,8,16,21,33,40,56,65,85,96,120,133,161,176,208,225,261,280,320,

%T 341,385,408,456,481,533,560,616,645,705,736,800,833,901,936,1008,

%U 1045,1121,1160,1240,1281,1365,1408,1496,1541,1633,1680,1776,1825,1925,1976

%N Generalized octagonal numbers: n*(3*n-2), n=0, +- 1, +- 2, +-3....

%C Numbers of the form 3*n^2+2*n, n an integer.

%C 3*a(n) + 1 is a perfect square.

%C a(n) mod 10 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. [Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 04 2009]

%C A089801 is the characteristic function. - _R. J. Mathar_, Oct 07 2011.

%C Exponents of powers of q in one form of the quintuple product identity. (-x^-2 + 1) * q^0 + (x^-3 - x) * q^1 + (-x^-5 + x^3) * q^5 + (x^-6 - x^4) * q^8 + ... = Sum_n q^(3*n^2 + 2*n) * (x^(3*n) - x^(-3*n - 2)) = Product_{k>0} (1 - x * q^(2*k - 1)) * (1 - x^-1 * q^(2*k - 1)) * (1 - q^(2*k)) * (1 - x^2 * q^(4*k)) * (1 - x^-2 * q^(4*k - 4)). - _Michael Somos_, Dec 21 2011

%C The offset 0 would also be valid here, all other entries of generalized k-gonal numbers have offset 0 (see cross references). - _Omar E. Pol_, Jan 12 2013

%C Also, x values of the Diophantine equation x(x+3)+(x+1)(x+2) = (x+y)^2+(x-y)^2. - _Bruno Berselli_, Mar 29 2013

%C Numbers n such that sum_{i=1..n} 2*i*(n-i)/n is an integer (the addend is the harmonic mean of i and n-i). - _Wesley Ivan Hurt_, Sep 14 2014

%C Equivalently, integers of the form m*(m+2)/3 (nonnegative values of m are listed in A032766). - _Bruno Berselli_, Jul 18 2016

%H T. D. Noe, <a href="/A001082/b001082.txt">Table of n, a(n) for n = 1..1000</a>

%H R. Stephan, <a href="http://www.ark.in-berlin.de/A001082.ps">On the solutions to 'px+1 is square'</a>

%H Z.-W. Sun, <a href="http://arxiv.org/abs/1503.03743">A result similar to Lagrange's theorem</a>, arXiv preprint arXiv:1503.03743 [math.NT], 2015.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/QuintupleProductIdentity.html">Quintuple Product Identity</a>

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

%F a(n) = n*(3*n-4)/4 if n even, (n-1)*(3*n+1)/4 if n odd.

%F a(n) = n^2 - n - floor(n/2)^2.

%F 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_, Aug 16 2002

%F a(n+1) = ceiling(n/2)^2 + A046092(floor(n/2)).

%F 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 O.g.f.: -x^2*(x^2+4*x+1)/((x-1)^3*(1+x)^2). - _R. J. Mathar_, Apr 15 2008

%F a(n) = n^2+n-ceiling(n/2)^2 with offset 0 and a(0)=0. [_Gary Detlefs_, Feb 23 2010]

%F a(n) = (6*n^2-6*n-1-(2*n-1)*(-1)^n)/8. - _Luce ETIENNE_, Dec 11 2014

%F E.g.f.: (3*x^2*exp(x) + x*exp(-x) - sinh(x))/4. - _Ilya Gutkovskiy_, Jul 15 2016

%F Sum_{n>=2} 1/a(n) = (9 + 2*sqrt(3)*Pi)/12. - _Vaclav Kotesovec_, Oct 05 2016

%e For the ninth comment: 65 is in the sequence because 65 = 13*(13+2)/3 or also 65 = -15*(-15+2)/3. - _Bruno Berselli_, Jul 18 2016

%p seq(n^2+n-ceil(n/2)^2, n=0..51); # _Gary Detlefs_, Feb 23 2010

%t Table[If[EvenQ[n], n*(3*n-4)/4, (n-1) (3*n+1)/4], {n, 100}]

%o (PARI) {a(n) = if( n%2, (n-1) * (3*n + 1) / 4, n * (3*n - 4) / 4)}

%o a001082 n = a001082_list !! n

%o a001082_list = scanl (+) 0 \$ tail a022998_list

%o -- _Reinhard Zumkeller_, Mar 31 2012

%o (MAGMA) [n^2 - n - Floor(n/2)^2 : n in [1..50]]; // _Wesley Ivan Hurt_, Sep 14 2014

%Y Partial sums of A022998.

%Y Cf. A000567, A005563, A085785, A089801, A245031.

%Y Column 4 of A195152. A045944.

%Y Generalized k-gonal numbers, k>=5: A001318, A000217, A085787, this sequence, A118277, A074377, A195160, A195162, A195313, A195818.

%Y Cf. sequences of the form m*(m+k)/(k+1) listed in A274978. [_Bruno Berselli_, Jul 25 2016]

%K nonn,easy

%O 1,3

%A _N. J. A. Sloane_ and _Tom Duff_

%E New sequence name from _Matthew Vandermast_, Apr 10 2003

%E Editorial changes by _N. J. A. Sloane_, Feb 03 2012

%E Edited by _Omar E. Pol_, Jun 09 2012

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