A091999 Numbers that are congruent to {2, 10} mod 12.

2, 10, 14, 22, 26, 34, 38, 46, 50, 58, 62, 70, 74, 82, 86, 94, 98, 106, 110, 118, 122, 130, 134, 142, 146, 154, 158, 166, 170, 178, 182, 190, 194, 202, 206, 214, 218, 226, 230, 238, 242, 250, 254, 262, 266, 274, 278, 286, 290, 298, 302, 310, 314, 322, 326, 334

Offset: 1

Author

Ray Chandler, Feb 21 2004

Keywords

• nonn
• easy

Comments

Numbers divisible by 2 but not by 3 or 4. - Robert Israel, Apr 24 2015

For n > 1, a(n) is representable as a sum of four but no fewer consecutive nonnegative integers, i.e., 10 = 1 + 2 + 3 + 4, 14 = 2 + 3 + 4 + 5, 22 = 4 + 5 + 6 + 7, etc. (see A138591). - Martin Renner, Mar 14 2016

Essentially the same as A063221. - Omar E. Pol, Aug 16 2023

Links

• David Lovler, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Second row of A092260.

Cf. A109761 (subsequence).

Cf. A002193, A007310, A101263, A138591, A186424.

Cf. A017545, A017641, A063221.

Programs

Haskell

a091999 n = a091999_list !! (n-1)
a091999_list = 2 : 10 : map (+ 12) a091999_list
-- Reinhard Zumkeller, Jan 21 2013

Magma

[6*n-3+(-1)^n : n in [1..100]]; // Wesley Ivan Hurt, Apr 23 2015

Maple

A091999:=n->6*n-3+(-1)^n: seq(A091999(n), n=1..100); # Wesley Ivan Hurt, Apr 23 2015

Mathematica

Flatten[#+{2,10}&/@(12*Range[0,30])] (* or *) LinearRecurrence[{1,1,-1},{2,10,14},60] (* Harvey P. Dale, Jun 24 2013 *)

PARI

a(n) = 6*n - 3 + (-1)^n \\ David Lovler, Jul 16 2022

Formula

a(n) = 2*A007310(n).

a(n) = A186424(n) - A186424(n-2), for n > 1.

a(n) = 12*(n-1) - a(n-1), with a(1)=2. - Vincenzo Librandi, Nov 16 2010

G.f.: 2*x*(1+4*x+x^2) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011

a(n) = a(n-1) + a(n-2) - a(n-3); a(1)=2, a(2)=10, a(3)=14. - Harvey P. Dale, Jun 24 2013

a(n) = 6*n - 3 + (-1)^n. - Wesley Ivan Hurt, Apr 23 2015

E.g.f.: 2 + (6*x - 2)*cosh(x) + 2*(3*x - 2)*sinh(x). - Stefano Spezia, May 09 2021

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(4*sqrt(3)). - Amiram Eldar, Dec 13 2021

E.g.f.: 2 + (6*x - 3)*exp(x) + exp(-x). - David Lovler, Aug 08 2022

a(n) = A063221(n), n > 1. - Omar E. Pol, Aug 15 2023

From Amiram Eldar, Nov 24 2024: (Start)

Product_{n>=1} (1 - (-1)^n/a(n)) = sqrt(2) (A002193).

Product_{n>=1} (1 + (-1)^n/a(n)) = 2*sin(Pi/12) (A101263). (End)