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A108118 Integers not divisible by 3 or 4. 3

%I #17 Sep 08 2022 08:45:19

%S 1,2,5,7,10,11,13,14,17,19,22,23,25,26,29,31,34,35,37,38,41,43,46,47,

%T 49,50,53,55,58,59,61,62,65,67,70,71,73,74,77,79,82,83,85,86,89,91,94,

%U 95,97,98,101,103,106,107,109,110,113,115,118,119,121,122,125,127,130,131

%N Integers not divisible by 3 or 4.

%C 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.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,-1,2,-1).

%F G.f.: x*(1+x^2)^2 / ( (1+x)*(x^2-x+1)*(x-1)^2 ). - _R. J. Mathar_, Oct 25 2011

%F From _Wesley Ivan Hurt_, Jul 22 2016: (Start)

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

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

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

%F 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)

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

%p 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

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

%o (Magma) [n : n in [0..150] | n mod 12 in [1, 2, 5, 7, 10, 11]]; // _Wesley Ivan Hurt_, Jul 22 2016

%Y Cf. A007310, A099477.

%K nonn,easy

%O 1,2

%A _Zak Seidov_, Jun 04 2005

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