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)).
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}]
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 5, 8, 16}, 60] (* Harvey P. Dale, Feb 03 2024 *)
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.
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]
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