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

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

Author

Wesley Ivan Hurt, May 30 2016

Keywords

• nonn
• easy

Comments

Numbers that are not congruent to {1, 2, 3, 4, 5, 6, 8, 9, 10, 11} mod 12.

Bisection of A083032.

Links

• David Lovler, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A008594, A017605, A083032, A049453, A049598.

Programs

Magma

[n : n in [0..400] | n mod 12 in [0, 7]];

Maple

A272975:=n->(12*n-11+(-1)^n)/2: seq(A272975(n), n=1..100);

Mathematica

Table[(12n - 11 + (-1)^n)/2, {n, 80}]

PARI

concat(0, Vec(x^2*(7+5*x)/((x-1)^2*(x+1)) + O(x^99))) \\ Altug Alkan, May 31 2016

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