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%I #92 Mar 10 2024 00:23:42
%S 11,13,26,28,41,43,56,58,71,73,86,88,101,103,116,118,131,133,146,148,
%T 161,163,176,178,191,193,206,208,221,223,236,238,251,253,266,268,281,
%U 283,296,298,311,313,326,328,341,343,356,358,371,373,386,388,401,403,416,418,431,433
%N Numbers congruent to 11 or 13 mod 15.
%C Includes every prime and twin prime (as pairs of consecutive primes) congruent to 11 or 13 mod 30.
%H Colin Barker, <a href="/A295340/b295340.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(n) = (1/4)*(-1)^n*(3*(-1)^n*(10*n + 1) - 11) for n > 0.
%F From _Colin Barker_, Dec 07 2017: (Start)
%F G.f.: x*(11 + 2*x + 2*x^2) / ((1 - x)^2*(1 + x)).
%F a(n) = (15*n - 4) / 2 for n even.
%F a(n) = (15*n + 7) / 2 for n odd.
%F a(n) = a(n-1) + a(n-2) - a(n-3) for n > 3.
%F (End)
%F a(n) = ceiling(15*n/2) + 5*(n mod 2) - 2 for n > 0. - _Mikk Heidemaa_, Sep 06 2018
%F a(n + 2) = a(n) + 15. - _David A. Corneth_, Sep 06 2018
%F a(n) = (11/2)*(n mod 2) + 15*n/2 - 2 for n > 0. - _Mikk Heidemaa_, Sep 08 2018
%F f(n) = 15*n - ((13*n + 17) mod 26) for n > 0 yields odd terms. - _Mikk Heidemaa_, Oct 28 2019
%F a(n) = 11*ceiling(1/2*n) + 2*n - 2 for n > 0. - _Mikk Heidemaa_, Nov 04 2019
%F E.g.f.: 2 + ((30*x + 3)*exp(x) - 11*exp(-x))/4. - _David Lovler_, Sep 08 2022
%t ParallelMap[11 * Ceiling[#/2] + 2 * # - 2 &, Range@ 10^3]
%t CoefficientList[ Series[(2x^2 + 2x + 11)/((1 + x) (x - 1)^2), {x, 0, 60}], x] (* or *)
%t LinearRecurrence[{1, 1, -1}, {11, 13, 26}, 60] (* _Robert G. Wilson v_, Jan 09 2018 *)
%t Select[Range[500], MemberQ[{11, 13}, Mod[#, 15]] &] (* _Vincenzo Librandi_, Sep 06 2018 *)
%t 11/2 * Mod[#, 2] + 15 * #/2 - 2 &/@ Range@ 500 (* _Mikk Heidemaa_, Sep 08 2018 *)
%o (PARI) Vec(x*(11 + 2*x + 2*x^2) / ((1 - x)^2*(1 + x)) + O(x^40)) \\ _Colin Barker_, Dec 07 2017
%o (PARI) a(n) = if(n%2, (15*n+7)/2, (15*n-4)/2); \\ _Altug Alkan_, Sep 06 2018
%o (PARI) a(n) = [11, -2][(n - 1)%2 + 1] + 15*(n \ 2) \\ _David A. Corneth_, Sep 06 2018
%o (Magma) [n: n in [1..500] | n mod 15 in [11, 13]]; // _Vincenzo Librandi_, Sep 06 2018
%Y Cf. A132238 (subsequence of primes), A132241 (subsequence of twin primes).
%K nonn,easy
%O 1,1
%A _Mikk Heidemaa_, Nov 20 2017
%E Name simplified by _David A. Corneth_, Sep 06 2018
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