A072833 -id:A072833 - cp's OEIS frontend

A108752 Numbers k such that 12 divides k*(k+1).

0, 3, 8, 11, 12, 15, 20, 23, 24, 27, 32, 35, 36, 39, 44, 47, 48, 51, 56, 59, 60, 63, 68, 71, 72, 75, 80, 83, 84, 87, 92, 95, 96, 99, 104, 107, 108, 111, 116, 119, 120, 123, 128, 131, 132, 135, 140, 143, 144, 147, 152, 155, 156, 159, 164, 167, 168, 171, 176, 179, 180

Offset: 1

Robert Phillips (bobp(AT)usca.edu), Jun 23 2005

First differences are 3, 5, 3, 1, 3, 5, 3, 1, 3, 5, 3, 1, 3, 5, 3, 1, ..., . - Robert G. Wilson v, May 31 2017

Numbers that are congruent to {0, 3, 8, 11} mod 12. - Amiram Eldar, Jul 26 2024

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Robert Phillips, Triangular numbers which are sums of two triangular numbers, 2006.
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Equals A112652-1, A218155-3, A174398-5, A072833+3.

Cf. A016777, A057077, A287765.

[3*n-2-(-1)^((2*n-3-(-1)^n) div 4): n in [1..80]]; // Vincenzo Librandi, May 04 2017

a:= proc(n) if is(n*(n+1)/12, integer) then n fi end: seq(a(n), n=0..200); # Emeric Deutsch, Jun 25 2005

Select[ Range[0, 182], Mod[ #(# + 1), 12] == 0 &] (* Robert G. Wilson v, Jun 25 2005 *)
LinearRecurrence[{2, -2, 2, -1}, {0, 3, 8, 11}, 200] (* Vincenzo Librandi, Jun 04 2017 *)

From R. J. Mathar, Jan 07 2009: (Start)
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) = A016777(n) - A057077(n).
G.f.: x*(3 + 2*x + x^2)/((1 + x^2)*(1 - x)^2). (End)
a(n) = 3*n - 2 - (-1)^((2*n-3-(-1)^n)/4). - Luce ETIENNE, Apr 04 2015
Sum_{n>=2} 1/a(n) = log(2)/2 + arccoth(sqrt(3))/(2*sqrt(3)) - Pi*(3+2*sqrt(3))/72. - Amiram Eldar, Jul 26 2024

More terms from Robert G. Wilson v and Emeric Deutsch, Jun 25 2005

A072835 Exponents occurring in expansion of F_9(q^2).

Original entry on oeis.org

0, 8, 14, 18, 20, 26, 32, 36, 38, 44, 50, 54, 56, 62, 68, 72, 74, 80, 86, 90, 92, 98, 104, 108, 110, 116, 122, 126, 128, 134, 140, 144, 146, 152, 158, 162, 164, 170, 176, 180, 182, 188, 194, 198, 200, 206, 212, 216, 218, 224, 230, 234, 236, 242, 248, 252, 254, 260, 266, 270, 272, 278

Offset: 0

N. J. A. Sloane, Jul 25 2002

Twice (A242660 without 1). Also, norms of vectors of the A*8 lattice. - _Andrey Zabolotskiy, Nov 10 2021

G. C. Greubel, Table of n, a(n) for n = 0..1000
S. Ahlgren, The sixth, eighth, ninth and tenth powers of Ramanujan's theta function, Proc. Amer. Math. Soc. 128 (2000), 1333-1338.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A072839, A242660.
Cf. A004014, A047222, A072833, A047328, A072834, A072836.
Cf. A056991.

f[x_, y_]:= QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y];
F[9,q_]:= f[q^9, q^9]^8 - 16*q^9*f[q^9, q^27]^8 + 256*q^18*f[q^18, q^54]^8 + 18*q^8*QPochhammer[q^18]^12/QPochhammer[q^6]^4;
cfs = CoefficientList[Series[F[9, q], {q, 0, 500}], q];
Take[Pick[Range[Length[cfs]] - 1, Sign[Abs[cfs]], 1], 50] (* G. C. Greubel, Apr 16 2018 *)

G.f.: -2*x*(x^4-x^3-2*x^2-3*x-4) / (x^5-x^4-x+1). - Colin Barker, Jul 31 2013
a(n+4) = a(n) + 18 for n > 0. - Jerzy R Borysowicz, Sep 02 2023
a(n)/n ~ 9/2. - Jerzy R Borysowicz, Sep 03 2023
a(n) = 2 * A056991(n+1) for n>=1. - Alois P. Heinz, Sep 03 2023

Terms a(22) onward added by G. C. Greubel, Apr 16 2018

cp's OEIS Frontend

A108752 Numbers k such that 12 divides k*(k+1).

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