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A272975 Numbers that are congruent to {0,7} mod 12.

%I A272975 #56 Sep 17 2023 10:00:22
%S A272975 0,7,12,19,24,31,36,43,48,55,60,67,72,79,84,91,96,103,108,115,120,127,
%T A272975 132,139,144,151,156,163,168,175,180,187,192,199,204,211,216,223,228,
%U A272975 235,240,247,252,259,264,271,276,283,288,295,300,307,312,319,324
%N A272975 Numbers that are congruent to {0,7} mod 12.
%C A272975 Numbers that are not congruent to {1, 2, 3, 4, 5, 6, 8, 9, 10, 11} mod 12.
%C A272975 Bisection of A083032.
%H A272975 David Lovler, <a href="/A272975/b272975.txt">Table of n, a(n) for n = 1..1000</a>
%H A272975 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F A272975 G.f.: x^2*(7+5*x) / ((x-1)^2*(x+1)).
%F A272975 a(n) = a(n-1) + a(n-2) - a(n-3) for n>3.
%F A272975 a(n) = (12*n - 11 + (-1)^n)/2.
%F A272975 a(2k) = A017605(k-1) k>0, a(2k-1) = A008594(k-1) k>0, a(2k)-a(2k-1) = 7.
%F A272975 a(n)-a(-n) = A008594(n) for n>0.
%F A272975 Sum_{i=1..n} a(2*i) = A049453(n) for n>0.
%F A272975 Sum_{i=1..n} a(2*i-1) = A049598(n-1) for n>0.
%F A272975 E.g.f.: 5 + ((12*x - 11)*exp(x) + exp(-x))/2. - _David Lovler_, Sep 04 2022
%F A272975 Sum_{n>=2} (-1)^n/a(n) = log(2)/4 + log(3)/8 - ((sqrt(3)-1)*Pi + 2*(sqrt(3)+3)*log(sqrt(3)+2))/(24*(sqrt(3)+1)). - _Amiram Eldar_, Sep 17 2023
%p A272975 A272975:=n->(12*n-11+(-1)^n)/2: seq(A272975(n), n=1..100);
%t A272975 Table[(12n - 11 + (-1)^n)/2, {n, 80}]
%o A272975 (Magma) [n : n in [0..400] | n mod 12 in [0, 7]];
%o A272975 (PARI) concat(0, Vec(x^2*(7+5*x)/((x-1)^2*(x+1)) + O(x^99))) \\ _Altug Alkan_, May 31 2016
%Y A272975 Cf. A008594, A017605, A083032, A049453, A049598.
%K A272975 nonn,easy
%O A272975 1,2
%A A272975 _Wesley Ivan Hurt_, May 30 2016