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Numbers that are congruent to {4, 5, 6} mod 7.
1

%I #18 Sep 08 2022 08:44:56

%S 4,5,6,11,12,13,18,19,20,25,26,27,32,33,34,39,40,41,46,47,48,53,54,55,

%T 60,61,62,67,68,69,74,75,76,81,82,83,88,89,90,95,96,97,102,103,104,

%U 109,110,111,116,117,118,123,124,125,130,131,132,137,138,139,144

%N Numbers that are congruent to {4, 5, 6} mod 7.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).

%F G.f.: x*(4+x+x^2+x^3) / ((1+x+x^2)*(x-1)^2). - _R. J. Mathar_, Oct 25 2011

%F From _Wesley Ivan Hurt_, Jun 07 2016: (Start)

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

%F a(n) = (21*n+3-12*cos(2*n*Pi/3)+4*sqrt(3)*sin(2*n*Pi/3))/9.

%F a(3k) = 7k-1, a(3k-1) = 7k-2, a(3k-2) = 7k-3. (End)

%p A047311:=n->(21*n+3-12*cos(2*n*Pi/3)+4*sqrt(3)*sin(2*n*Pi/3))/9: seq(A047311(n), n=1..100); # _Wesley Ivan Hurt_, Jun 07 2016

%t LinearRecurrence[{1,0,1,-1}, {4,5,6,11}, 60] (* _Harvey P. Dale_, Oct 29 2014 *)

%o (Magma) [n : n in [0..150] | n mod 7 in [4..6]]; // _Wesley Ivan Hurt_, Jun 07 2016

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_