A108118 Integers not divisible by 3 or 4.

1, 2, 5, 7, 10, 11, 13, 14, 17, 19, 22, 23, 25, 26, 29, 31, 34, 35, 37, 38, 41, 43, 46, 47, 49, 50, 53, 55, 58, 59, 61, 62, 65, 67, 70, 71, 73, 74, 77, 79, 82, 83, 85, 86, 89, 91, 94, 95, 97, 98, 101, 103, 106, 107, 109, 110, 113, 115, 118, 119, 121, 122, 125, 127, 130, 131

Offset: 1

Author

Zak Seidov, Jun 04 2005

Keywords

• nonn
• easy

Comments

Or, numbers congruent to {1, 2, 5, 7, 10, 11} mod 12 (cf. A007310). Expand (x+x^2+x^5+x^7+x^10+x^11)/(1-x^12) (cf. A007310). All terms, except 35 and 70, are also in A099477.

Links

• Index entries for linear recurrences with constant coefficients, signature (2,-1,-1,2,-1).

Crossrefs

Cf. A007310, A099477.

Programs

Magma

[n : n in [0..150] | n mod 12 in [1, 2, 5, 7, 10, 11]]; // Wesley Ivan Hurt, Jul 22 2016

Maple

A108118:=n->12*floor(n/6)+[1, 2, 5, 7, 10, 11][(n mod 6)+1]: seq(A108118(n), n=0..100); # Wesley Ivan Hurt, Jul 22 2016

Mathematica

Select[ Range[132], !IntegerQ[ #/4] && !IntegerQ[ #/3] &] (* or *) Flatten[ NestList[12 + # &, {1, 2, 5, 7, 10, 11}, 10]]

Formula

G.f.: x*(1+x^2)^2 / ( (1+x)*(x^2-x+1)*(x-1)^2 ). - R. J. Mathar, Oct 25 2011

From Wesley Ivan Hurt, Jul 22 2016: (Start)

a(n) = 2*a(n-1) - a(n-2) - a(n-3) + 2*a(n-4) - a(n-5) for n>5.

a(n) = a(n-6) + 12 for n>6.

a(n) = (6*n - 3 + cos(n*Pi/3) - cos(n*Pi) - sqrt(3)*sin(n*Pi/3))/3.

a(6k) = 12k-1, a(6k-1) = 12k-2, a(6k-2) = 12k-5, a(6k-3) = 12k-7, a(6k-4) = 12k-10, a(6k-5) = 12k-11. (End)

Sum_{n>=1} (-1)^(n+1)/a(n) = (4-sqrt(3))*Pi/12. - Amiram Eldar, Jan 01 2022