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

1,1

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).

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)

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 *)

Prog

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

Crossrefs

Second row of A092260.

Cf. A109761 (subsequence).

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

Cf. A017545, A017641, A063221.

Sequence in context: A032624 A350059 A349833 * A230624 A290143 A160773

Adjacent sequences: A091996 A091997 A091998 * A092000 A092001 A092002

Keyword

nonn,easy

Author

Ray Chandler, Feb 21 2004

Status

approved