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A047377 Numbers that are congruent to {0, 1, 4, 5} mod 7.

%I A047377 #29 Sep 08 2022 08:44:57
%S A047377 0,1,4,5,7,8,11,12,14,15,18,19,21,22,25,26,28,29,32,33,35,36,39,40,42,
%T A047377 43,46,47,49,50,53,54,56,57,60,61,63,64,67,68,70,71,74,75,77,78,81,82,
%U A047377 84,85,88,89,91,92,95,96,98,99,102,103,105,106,109,110
%N A047377 Numbers that are congruent to {0, 1, 4, 5} mod 7.
%H A047377 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F A047377 a(n+1) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=1, b(1)=4 and b(k)=7*2^(k-2) for k>1. - _Philippe Deléham_, Oct 25 2011
%F A047377 G.f.: x^2*(1+3*x+x^2+2*x^3) / ( (1+x)*(x^2+1)*(x-1)^2 ). - _R. J. Mathar_, Dec 04 2011
%F A047377 From _Wesley Ivan Hurt_, May 24 2016: (Start)
%F A047377 a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
%F A047377 a(n) = (14*n-15-3*i^(2*n)+(1-i)*i^(-n)+(1+i)*i^n)/8, where i=sqrt(-1).
%F A047377 a(2k) = A047383(k), a(2k-1) = A047345(k). (End)
%F A047377 E.g.f.: (8 - sin(x) + cos(x) + (7*x - 6)*sinh(x) + (7*x - 9)*cosh(x))/4. - _Ilya Gutkovskiy_, May 25 2016
%p A047377 A047377:=n->(14*n-15-3*I^(2*n)+(1-I)*I^(-n)+(1+I)*I^n)/8: seq(A047377(n), n=1..100); # _Wesley Ivan Hurt_, May 24 2016
%t A047377 Table[(14n-15-3*I^(2n)+(1-I)*I^(-n)+(1+I)*I^n)/8, {n, 80}] (* _Wesley Ivan Hurt_, May 24 2016 *)
%t A047377 Select[Range@ 120, MemberQ[{0, 1, 4, 5}, Mod[#, 7]] &] (* _Michael De Vlieger_, May 24 2016 *)
%t A047377 a[n_] :=  n + Floor[(n - 1)/2] +  Floor[(n - 3)/4];
%t A047377 Table[a[n], {n, 1, 64}] (* _Peter Luschny_, Dec 23 2021 *)
%o A047377 (Magma) [n : n in [0..150] | n mod 7 in [0, 1, 4, 5]]; // _Wesley Ivan Hurt_, May 24 2016
%Y A047377 Cf. A030308, A047345, A047383.
%K A047377 nonn,easy
%O A047377 1,3
%A A047377 _N. J. A. Sloane_
%E A047377 More terms from _Wesley Ivan Hurt_, May 24 2016