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

Author

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

Keywords

• nonn
• easy

Comments

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

Links

• 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).

Crossrefs

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

Cf. A016777, A057077, A287765.

Programs

Magma

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

Maple

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

Mathematica

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 *)

Formula

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

Extensions

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