A091999 - OEIS

%I #75 Nov 24 2024 01:51:20

%S 2,10,14,22,26,34,38,46,50,58,62,70,74,82,86,94,98,106,110,118,122,

%T 130,134,142,146,154,158,166,170,178,182,190,194,202,206,214,218,226,

%U 230,238,242,250,254,262,266,274,278,286,290,298,302,310,314,322,326,334

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

%C Numbers divisible by 2 but not by 3 or 4. - _Robert Israel_, Apr 24 2015

%C 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

%C Essentially the same as A063221. - _Omar E. Pol_, Aug 16 2023

%H David Lovler, <a href="/A091999/b091999.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).

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

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

%F a(n) = 12*(n-1) - a(n-1), with a(1)=2. - _Vincenzo Librandi_, Nov 16 2010

%F G.f.: 2*x*(1+4*x+x^2) / ( (1+x)*(x-1)^2 ). - _R. J. Mathar_, Oct 08 2011

%F 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

%F a(n) = 6*n - 3 + (-1)^n. - _Wesley Ivan Hurt_, Apr 23 2015

%F E.g.f.: 2 + (6*x - 2)*cosh(x) + 2*(3*x - 2)*sinh(x). - _Stefano Spezia_, May 09 2021

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

%F E.g.f.: 2 + (6*x - 3)*exp(x) + exp(-x). - _David Lovler_, Aug 08 2022

%F a(n) = A063221(n), n > 1. - _Omar E. Pol_, Aug 15 2023

%F From _Amiram Eldar_, Nov 24 2024: (Start)

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

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

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

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

%o (Haskell)

%o a091999 n = a091999_list !! (n-1)

%o a091999_list = 2 : 10 : map (+ 12) a091999_list

%o -- _Reinhard Zumkeller_, Jan 21 2013

%o (Magma) [6*n-3+(-1)^n : n in [1..100]]; // _Wesley Ivan Hurt_, Apr 23 2015

%o (PARI) a(n) = 6*n - 3 + (-1)^n \\ _David Lovler_, Jul 16 2022

%Y Second row of A092260.

%Y Cf. A109761 (subsequence).

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

%Y Cf. A017545, A017641, A063221.

%K nonn,easy

%O 1,1

%A _Ray Chandler_, Feb 21 2004