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Numbers that are congruent to {0, 4, 5, 6} mod 7.
1

%I #17 Sep 08 2022 08:44:56

%S 0,4,5,6,7,11,12,13,14,18,19,20,21,25,26,27,28,32,33,34,35,39,40,41,

%T 42,46,47,48,49,53,54,55,56,60,61,62,63,67,68,69,70,74,75,76,77,81,82,

%U 83,84,88,89,90,91,95,96,97,98,102,103,104,105,109,110,111

%N Numbers that are congruent to {0, 4, 5, 6} mod 7.

%H Daniel Starodubtsev, <a href="/A047312/b047312.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).

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

%F From _Wesley Ivan Hurt_, Jun 03 2016: (Start)

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

%F a(n) = (14*n-5+3*i^(2*n)-(3+3*i)*i^(-n)-(3-3*i)*i^n)/8 where i=sqrt(-1).

%F a(2k) = A047288(k), a(2k-1) = A047382(k). (End)

%p A047312:=n->(14*n-5+3*I^(2*n)-(3+3*I)*I^(-n)-(3-3*I)*I^n)/8: seq(A047312(n), n=1..100); # _Wesley Ivan Hurt_, Jun 03 2016

%t Table[(14n-5+3*I^(2n)-(3+3*I)*I^(-n)-(3-3*I)*I^n)/8, {n, 80}] (* _Wesley Ivan Hurt_, Jun 03 2016 *)

%o (Magma) [n : n in [0..150] | n mod 7 in [0, 4, 5, 6]]; // _Wesley Ivan Hurt_, Jun 03 2016

%Y Cf. A047288, A047382.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_