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

0, 1, 5, 8, 16, 21, 33, 40, 56, 65, 85, 96, 120, 133, 161, 176, 208, 225, 261, 280, 320, 341, 385, 408, 456, 481, 533, 560, 616, 645, 705, 736, 800, 833, 901, 936, 1008, 1045, 1121, 1160, 1240, 1281, 1365, 1408, 1496, 1541, 1633, 1680, 1776, 1825, 1925, 1976

Offset: 1

Author

N. J. A. Sloane and Tom Duff

Keywords

• nonn
• easy

Comments

Numbers of the form 3*m^2+2*m, m an integer.

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

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, Sep 04 2009

A089801 is the characteristic function. - R. J. Mathar, Oct 07 2011

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>=0} 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

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

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

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

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

Exponents of q in the expansion of Sum_{n >= 0} ( q^n * Product_{k = 1..n} (1 - q^(2*k-1)) ) = 1 + q - q^5 - q^8 + q^16 + q^21 - - + + .... - Peter Bala, Dec 03 2020

Exponents of q in the expansion of Product_{n >= 1} (1 - q^(6*n))*(1 + q^(6*n-1))*(1 + q^(6*n-5)) = 1 + q + q^5 + q^8 + q^16 + q^21 + .... - Peter Bala, Dec 09 2020

Exponents of q in the expansion of Product_{n >= 1} (1 - q^n)^2*(1 - q^(4*n))^2 /(1 - q^(2*n)) = 1 - 2*q + 4*q^5 - 5*q^8 + 7*q^16 - + ... (a consequence of the quintuple product identity). The series coefficients are a signed version of A001651. - Peter Bala, Feb 16 2021

From Peter Bala, Nov 26 2024: (Start)

Apart from the first two terms, the exponents of q in the expansion of Sum_{n >= 1} q^(3*n+2) * (Product_{k = 2..n} 1 - q^(2*k-1)) = q^5 + q^8 - q^16 - q^21 + + - - ... (in Andrews, equation 8, replace q with q^2 and set x = q).

Exponents of q^2 in the expansion of Sum_{n >= 0} q^n / (Product_{k = 1..n+1 } 1 + q^(2*k-1)) = 1 + (q^2)^1 - (q^2)^5 - (q^2)^8 + (q^2)^16 + (q^2)^21 - - + + ... (Chen, equation 22). (End)

Examples

			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
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• George E. Andrews, Euler's Pentagonal Number Theorem, Mathematics Magazine, Vol. 56, No. 5 (Nov., 1983), pp. 279-284.
• Sandy H. L. Chen, Symmetric Identities from an Invariant in Partition Conjugation and Their Applications in q-Series, Open Journal of Discrete Mathematics, Vol.4, No.2, April 2014.
• John Elias, Illustration of initial terms: Hourglass Octagonals.
• John Elias, Illustration: Pentagonal-Octagonal Star Configurations
• Ralf Stephan, On the solutions to 'px+1 is square'.
• Zhi-Wei Sun, A result similar to Lagrange's theorem, arXiv preprint arXiv:1503.03743 [math.NT], 2015.
• Eric Weisstein's World of Mathematics, Quintuple Product Identity.
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Partial sums of A022998.

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

Column 4 of A195152. A045944.

Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), this sequence (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).

Cf. sequences of the form m*(m+k)/(k+1) listed in A274978. [Bruno Berselli, Jul 25 2016]

Programs

Haskell

a001082 n = a001082_list !! n
a001082_list = scanl (+) 0 $ tail a022998_list
-- Reinhard Zumkeller, Mar 31 2012

Magma

[n^2 - n - Floor(n/2)^2 : n in [1..50]]; // Wesley Ivan Hurt, Sep 14 2014

Maple

seq(n*(n-1)-floor(n/2)^2, n=1..51); # Gary Detlefs, Feb 23 2010

Mathematica

Table[If[EvenQ[n], n*(3*n-4)/4, (n-1) (3*n+1)/4], {n, 100}]
LinearRecurrence[{1,2,-2,-1,1},{0,1,5,8,16},60] (* Harvey P. Dale, Feb 03 2024 *)

PARI

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

Formula

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

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

G.f.: Sum_{n>=0} (-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

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

a(2n) = n(3n-2) = A000567(n), a(2n+1) = n(3n+2) = A045944(n). - Mohamed Bouhamida, Nov 06 2007

O.g.f.: -x^2*(x^2+4*x+1)/((x-1)^3*(1+x)^2). - R. J. Mathar, Apr 15 2008

a(n) = n^2+n-ceiling(n/2)^2 with offset 0 and a(0)=0. - Gary Detlefs, Feb 23 2010

a(n) = (6*n^2-6*n-1-(2*n-1)*(-1)^n)/8. - Luce ETIENNE, Dec 11 2014

E.g.f.: (3*x^2*exp(x) + x*exp(-x) - sinh(x))/4. - Ilya Gutkovskiy, Jul 15 2016

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

Sum_{n>=2} (-1)^n/a(n) = 3*log(3)/2 - 3/4. - Amiram Eldar, Feb 28 2022

Extensions

New sequence name from Matthew Vandermast, Apr 10 2003

Editorial changes by N. J. A. Sloane, Feb 03 2012

Edited by Omar E. Pol, Jun 09 2012