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A272975
Numbers that are congruent to {0,7} mod 12.
2
0, 7, 12, 19, 24, 31, 36, 43, 48, 55, 60, 67, 72, 79, 84, 91, 96, 103, 108, 115, 120, 127, 132, 139, 144, 151, 156, 163, 168, 175, 180, 187, 192, 199, 204, 211, 216, 223, 228, 235, 240, 247, 252, 259, 264, 271, 276, 283, 288, 295, 300, 307, 312, 319, 324
OFFSET
1,2
COMMENTS
Numbers that are not congruent to {1, 2, 3, 4, 5, 6, 8, 9, 10, 11} mod 12.
Bisection of A083032.
FORMULA
G.f.: x^2*(7+5*x) / ((x-1)^2*(x+1)).
a(n) = a(n-1) + a(n-2) - a(n-3) for n>3.
a(n) = (12*n - 11 + (-1)^n)/2.
a(2k) = A017605(k-1) k>0, a(2k-1) = A008594(k-1) k>0, a(2k)-a(2k-1) = 7.
a(n)-a(-n) = A008594(n) for n>0.
Sum_{i=1..n} a(2*i) = A049453(n) for n>0.
Sum_{i=1..n} a(2*i-1) = A049598(n-1) for n>0.
E.g.f.: 5 + ((12*x - 11)*exp(x) + exp(-x))/2. - David Lovler, Sep 04 2022
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
MAPLE
A272975:=n->(12*n-11+(-1)^n)/2: seq(A272975(n), n=1..100);
MATHEMATICA
Table[(12n - 11 + (-1)^n)/2, {n, 80}]
PROG
(Magma) [n : n in [0..400] | n mod 12 in [0, 7]];
(PARI) concat(0, Vec(x^2*(7+5*x)/((x-1)^2*(x+1)) + O(x^99))) \\ Altug Alkan, May 31 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, May 30 2016
STATUS
approved