A092259 Numbers that are congruent to {4, 8} mod 12.
4, 8, 16, 20, 28, 32, 40, 44, 52, 56, 64, 68, 76, 80, 88, 92, 100, 104, 112, 116, 124, 128, 136, 140, 148, 152, 160, 164, 172, 176, 184, 188, 196, 200, 208, 212, 220, 224, 232, 236, 244, 248, 256, 260, 268, 272, 280, 284, 292, 296, 304, 308, 316, 320, 328, 332
Offset: 1
Links
- Index entries for linear recurrences with constant coefficients, signature (1,1,-1).
Programs
-
Magma
[n : n in [0..400] | n mod 12 in [4, 8]]; // Wesley Ivan Hurt, May 21 2016
-
Maple
A092259:=n->6*n-3-(-1)^n: seq(A092259(n), n=1..100); # Wesley Ivan Hurt, May 21 2016
-
Mathematica
Table[6n-3-(-1)^n, {n, 80}] (* Wesley Ivan Hurt, May 21 2016 *) LinearRecurrence[{1,1,-1},{4,8,16},60] (* Harvey P. Dale, Oct 07 2021 *)
Formula
G.f.: 4*x*(1+x+x^2) / ( (1+x)*(x-1)^2 ).
a(n) = 4 * A001651(n).
Iff phi(n) = phi(3n/2), then n is in A069587. - Labos Elemer, Feb 25 2004
a(n) = 12*(n-1)-a(n-1) (with a(1)=4). - Vincenzo Librandi, Nov 16 2010
From Wesley Ivan Hurt, May 21 2016: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3) for n>3.
a(n) = 6n - 3 - (-1)^n.
a(n) = -a(1-n), a(n) = A092899(n) + 1 for n>0. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi*sqrt(3)/36. - Amiram Eldar, Dec 30 2021
From Amiram Eldar, Nov 24 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 1/sqrt(2) + 1/sqrt(6) (A145439).
Product_{n>=1} (1 + (-1)^n/a(n)) = sqrt(2/3) (A157697). (End)
Extensions
Edited and extended by Ray Chandler, Feb 21 2004