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%I #39 Sep 26 2022 05:40:46
%S 1,3,13,15,25,27,37,39,49,51,61,63,73,75,85,87,97,99,109,111,121,123,
%T 133,135,145,147,157,159,169,171,181,183,193,195,205,207,217,219,229,
%U 231,241,243,253,255,265,267,277,279,289,291,301,303,313,315,325,327
%N Numbers congruent to {1, 3} mod 12.
%C Positive integers k such that Hypergeometric[k/4,(4-k)/4,1/2,3/4] = 2*cos(Pi/6).
%H David Lovler, <a href="/A146512/b146512.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F a(2k-1) = 12*(k-1)+1, a(2k) = 12*(k-1)+3, where k>0.
%F With offset 0, a(n) = 8*floor(n/2) + 2*n + 1, or a(n) = 6*n - 1 + 2*(-1)^n. - _Gary Detlefs_, Mar 13 2010
%F a(n) = 12*n-a(n-1)-20 (with a(1)=1). - _Vincenzo Librandi_, Nov 26 2010
%F G.f.: x * (1 + 2*x + 9*x^2) / (1 - x - x^2 + x^3). - _Michael Somos_, Dec 06 2016
%F a(n) = a(n-1)+a(n-2)-a(n-3). - _Wesley Ivan Hurt_, May 03 2021
%F E.g.f.: 9 + (6*x - 7)*exp(x) - 2*exp(-x). - _David Lovler_, Sep 07 2022
%F Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(3)+1)*(2*Pi + 2*arccosh(26) - 4*sqrt(3)*arccoth(sqrt(3)) + 3*(sqrt(3)-1)*log(3))/48. - _Amiram Eldar_, Sep 26 2022
%e G.f. = x + 3*x^2 + 13*x^3 + 15*x^4 + 25*x^5 + 27*x^6 + 37*x^7 + 39*x^8 + ...
%t Select[Range[300],MemberQ[{1,3},Mod[#,12]]&] (* _Ray Chandler_, Dec 06 2016 *)
%o (PARI) {a(n) = 6*n - 9 + n%2*4}; /* _Michael Somos_, Dec 06 2016 */
%Y Cf. A146507, A146509, A146510, A146511.
%K nonn,easy
%O 1,2
%A _Artur Jasinski_, Oct 30 2008
%E Formula and crossrefs corrected by _Ray Chandler_, Dec 06 2016
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