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%I #31 Sep 08 2022 08:44:57
%S 0,1,6,7,8,9,14,15,16,17,22,23,24,25,30,31,32,33,38,39,40,41,46,47,48,
%T 49,54,55,56,57,62,63,64,65,70,71,72,73,78,79,80,81,86,87,88,89,94,95,
%U 96,97,102,103,104,105,110,111,112,113,118,119,120,121,126
%N Numbers that are congruent to {0, 1, 6, 7} mod 8.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F a(n+1) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=1, b(1)=6 and b(k)=2^(k+1) for k>1. - _Philippe Deléham_, Oct 19 2011
%F a(n) = 2n - A010873(n+1). - _Wesley Ivan Hurt_, Jul 07 2013
%F G.f.: x^2*(1+5*x+x^2+x^3) / ( (1+x)*(1+x^2)*(x-1)^2 ). - _R. J. Mathar_, Jul 14 2013
%F From _Wesley Ivan Hurt_, May 29 2016: (Start)
%F a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
%F a(n) = (4*n-3-i^(2*n)+(1-i)*i^(-n)+(1+i)*i^n)/2 where i=sqrt(-1).
%F a(2k) = A047522(k), a(2k-1) = A047451(k). (End)
%F E.g.f.: 1 - sin(x) + cos(x) + (2*x - 1)*sinh(x) + 2*(x - 1)*cosh(x). - _Ilya Gutkovskiy_, May 29 2016
%F Sum_{n>=2} (-1)^n/a(n) = Pi/16 + (5-sqrt(2))*log(2)/8 + sqrt(2)*log(2+sqrt(2))/4. - _Amiram Eldar_, Dec 20 2021
%p A047551:=n->(4*n-3-I^(2*n)+(1-I)*I^(-n)+(1+I)*I^n)/2: seq(A047551(n), n=1..100); # _Wesley Ivan Hurt_, May 29 2016
%t Table[(4n-3-I^(2n)+(1-I)*I^(-n)+(1+I)*I^n)/2, {n, 80}] (* _Wesley Ivan Hurt_, May 29 2016 *)
%o (Magma) [n : n in [0..150] | n mod 8 in [0, 1, 6, 7]]; // _Wesley Ivan Hurt_, May 29 2016
%Y Cf. A010873, A030308, A047451, A047522.
%K nonn,easy
%O 1,3
%A _N. J. A. Sloane_
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