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 A001082 Generalized octagonal numbers: k*(3*k-2), k=0, +- 1, +- 2, +-3, ... 123
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 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 identity 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 LINKS T. D. Noe, Table of n, a(n) for n = 1..1000 John Elias, Illustration of initial terms: Hourglass Octagonals. 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). 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 EXAMPLE 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 MAPLE seq(n^2+n-ceil(n/2)^2, n=0..51); # Gary Detlefs, Feb 23 2010 MATHEMATICA Table[If[EvenQ[n], n*(3*n-4)/4, (n-1) (3*n+1)/4], {n, 100}] PROG (PARI) {a(n) = if( n%2, (n-1) * (3*n + 1) / 4, n * (3*n - 4) / 4)}; (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 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] Sequence in context: A314564 A314565 A126695 * A242090 A030006 A229849 Adjacent sequences: A001079 A001080 A001081 * A001083 A001084 A001085 KEYWORD nonn,easy AUTHOR 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 STATUS approved

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Last modified December 10 02:09 EST 2022. Contains 358712 sequences. (Running on oeis4.)