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A047606 Numbers that are congruent to {1, 2, 3, 5} mod 8.

%I A047606 #24 Sep 08 2022 08:44:57
%S A047606 1,2,3,5,9,10,11,13,17,18,19,21,25,26,27,29,33,34,35,37,41,42,43,45,
%T A047606 49,50,51,53,57,58,59,61,65,66,67,69,73,74,75,77,81,82,83,85,89,90,91,
%U A047606 93,97,98,99,101,105,106,107,109,113,114,115,117,121,122,123
%N A047606 Numbers that are congruent to {1, 2, 3, 5} mod 8.
%H A047606 Bruno Berselli, <a href="/A047606/b047606.txt">Table of n, a(n) for n = 1..1000</a>
%H A047606 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F A047606 From _Bruno Berselli_, Jul 17 2012: (Start)
%F A047606 G.f.: x*(1+x+x^2+2*x^3+3*x^4)/((1+x)*(1-x)^2*(1+x^2)).
%F A047606 a(n) = 2*n-3+(3-(-1)^n)*(1-i^(n*(n+1)))/4, where i=sqrt(-1). (End)
%F A047606 From _Wesley Ivan Hurt_, Jun 02 2016: (Start)
%F A047606 a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
%F A047606 a(2k) = A047617(k), a(2k-1) = A047471(k). (End)
%F A047606 E.g.f.: (6 + 2*sin(x) - cos(x) + 4*(x - 1)*sinh(x) + (4*x - 5)*cosh(x))/2. - _Ilya Gutkovskiy_, Jun 03 2016
%F A047606 Sum_{n>=1} (-1)^(n+1)/a(n) = (3*sqrt(2)-2)*Pi/16 + (2-sqrt(2))*log(2)/16 + sqrt(2)*log(2+sqrt(2))/8. - _Amiram Eldar_, Dec 23 2021
%p A047606 A047606:=n->2*n-3+(3-I^(2*n))*(1-I^(n*(n+1)))/4: seq(A047606(n), n=1..100); # _Wesley Ivan Hurt_, Jun 02 2016
%t A047606 Select[Range[120], MemberQ[{1, 2, 3, 5}, Mod[#, 8]] &] (* or *) LinearRecurrence[{1, 0, 0, 1, -1}, {1, 2, 3, 5, 9}, 60] (* _Bruno Berselli_, Jul 17 2012 *)
%o A047606 From _Bruno Berselli_, Jul 17 2012: (Start)
%o A047606 (Magma) [n: n in [1..120] | n mod 8 in [1,2,3,5]];
%o A047606 (Maxima) makelist(2*n-3+(3-(-1)^n)*(1-%i^(n*(n+1)))/4,n,1,60);
%o A047606 (PARI) Vec((1+x+x^2+2*x^3+3*x^4)/((1+x)*(1-x)^2*(1+x^2))+O(x^60)) (End)
%Y A047606 Cf. A047471, A047617.
%K A047606 nonn,easy
%O A047606 1,2
%A A047606 _N. J. A. Sloane_