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

%I #24 Sep 08 2022 08:44:57

%S 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 49,50,51,53,57,58,59,61,65,66,67,69,73,74,75,77,81,82,83,85,89,90,91,

%U 93,97,98,99,101,105,106,107,109,113,114,115,117,121,122,123

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

%H Bruno Berselli, <a href="/A047606/b047606.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 _Bruno Berselli_, Jul 17 2012: (Start)

%F G.f.: x*(1+x+x^2+2*x^3+3*x^4)/((1+x)*(1-x)^2*(1+x^2)).

%F a(n) = 2*n-3+(3-(-1)^n)*(1-i^(n*(n+1)))/4, where i=sqrt(-1). (End)

%F From _Wesley Ivan Hurt_, Jun 02 2016: (Start)

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

%F a(2k) = A047617(k), a(2k-1) = A047471(k). (End)

%F 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 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:=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 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 From _Bruno Berselli_, Jul 17 2012: (Start)

%o (Magma) [n: n in [1..120] | n mod 8 in [1,2,3,5]];

%o (Maxima) makelist(2*n-3+(3-(-1)^n)*(1-%i^(n*(n+1)))/4,n,1,60);

%o (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 Cf. A047471, A047617.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_