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A047624 Numbers that are congruent to {0, 1, 3, 5} mod 8.

%I A047624 #26 Sep 08 2022 08:44:57
%S A047624 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 A047624 49,51,53,56,57,59,61,64,65,67,69,72,73,75,77,80,81,83,85,88,89,91,93,
%U A047624 96,97,99,101,104,105,107,109,112,113,115,117,120,121,123
%N A047624 Numbers that are congruent to {0, 1, 3, 5} mod 8.
%H A047624 G. C. Greubel, <a href="/A047624/b047624.txt">Table of n, a(n) for n = 1..1000</a>
%H A047624 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F A047624 From _Reinhard Zumkeller_, Feb 21 2010: (Start)
%F A047624 a(n+1) = A173562(n) - A173562(n-1);
%F A047624 a(n+1) - a(n) = A140081(n-1) + 1;
%F A047624 a(n) = A140201(n-1) + A042948(A004526(n-1)). (End)
%F A047624 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 A047624 From _Wesley Ivan Hurt_, Jun 01 2016: (Start)
%F A047624 a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
%F A047624 a(n) = (8*n-11-i^(2*n)+i^(1-n)-i^(1+n))/4 where i=sqrt(-1).
%F A047624 a(2k) = A016813(k-1) for k>0, a(2k-1) = A047470(k). (End)
%F A047624 E.g.f.: (6 + sin(x) + (4*x - 5)*sinh(x) + (4*x - 6)*cosh(x))/2. - _Ilya Gutkovskiy_, Jun 01 2016
%F A047624 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 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 A047624 Table[(8n-11-I^(2n)+I^(1-n)-I^(1+n))/4, {n, 80}] (* _Wesley Ivan Hurt_, Jun 01 2016 *)
%t A047624 LinearRecurrence[{1,0,0,1,-1},{0,1,3,5,8},100] (* _G. C. Greubel_, Jun 01 2016 *)
%o A047624 (Magma) [n : n in [0..150] | n mod 8 in [0, 1, 3, 5]]; // _Wesley Ivan Hurt_, Jun 01 2016
%Y A047624 Cf. A004526, A016813, A042948, A047470, A140081, A140201, A173562.
%K A047624 nonn,easy
%O A047624 1,3
%A A047624 _N. J. A. Sloane_