%I #26 Sep 08 2022 08:44:57

%S 0,1,3,5,8,9,11,13,16,17,19,21,24,25,27,29,32,33,35,37,40,41,43,45,48,

%T 49,51,53,56,57,59,61,64,65,67,69,72,73,75,77,80,81,83,85,88,89,91,93,

%U 96,97,99,101,104,105,107,109,112,113,115,117,120,121,123

%N Numbers that are congruent to {0, 1, 3, 5} mod 8.

%H G. C. Greubel, <a href="/A047624/b047624.txt">Table of n, a(n) for n = 1..1000</a>

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

%F From _Reinhard Zumkeller_, Feb 21 2010: (Start)

%F a(n+1) = A173562(n) - A173562(n-1);

%F a(n+1) - a(n) = A140081(n-1) + 1;

%F a(n) = A140201(n-1) + A042948(A004526(n-1)). (End)

%F G.f.: x^2*(1+2*x+2*x^2+3*x^3) / ( (1+x)*(x^2+1)*(x-1)^2 ). - _R. J. Mathar_, Oct 08 2011

%F From _Wesley Ivan Hurt_, Jun 01 2016: (Start)

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

%F a(n) = (8*n-11-i^(2*n)+i^(1-n)-i^(1+n))/4 where i=sqrt(-1).

%F a(2k) = A016813(k-1) for k>0, a(2k-1) = A047470(k). (End)

%F E.g.f.: (6 + sin(x) + (4*x - 5)*sinh(x) + (4*x - 6)*cosh(x))/2. - _Ilya Gutkovskiy_, Jun 01 2016

%F Sum_{n>=2} (-1)^n/a(n) = (3-sqrt(2))*Pi/16 + (8-sqrt(2))*log(2)/16 + sqrt(2)*log(2+sqrt(2))/8. - _Amiram Eldar_, Dec 20 2021

%p A047624:=n->(8*n-11-I^(2*n)+I^(1-n)-I^(1+n))/4: seq(A047624(n), n=1..100); # _Wesley Ivan Hurt_, Jun 01 2016

%t Table[(8n-11-I^(2n)+I^(1-n)-I^(1+n))/4, {n, 80}] (* _Wesley Ivan Hurt_, Jun 01 2016 *)

%t LinearRecurrence[{1,0,0,1,-1},{0,1,3,5,8},100] (* _G. C. Greubel_, Jun 01 2016 *)

%o (Magma) [n : n in [0..150] | n mod 8 in [0, 1, 3, 5]]; // _Wesley Ivan Hurt_, Jun 01 2016

%Y Cf. A004526, A016813, A042948, A047470, A140081, A140201, A173562.

%K nonn,easy

%O 1,3

%A _N. J. A. Sloane_