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Numbers that are congruent to {1, 3, 4, 5} mod 8.
1

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

%S 1,3,4,5,9,11,12,13,17,19,20,21,25,27,28,29,33,35,36,37,41,43,44,45,

%T 49,51,52,53,57,59,60,61,65,67,68,69,73,75,76,77,81,83,84,85,89,91,92,

%U 93,97,99,100,101,105,107,108,109,113,115,116,117,121,123,124

%N Numbers that are congruent to {1, 3, 4, 5} mod 8.

%H Bruno Berselli, <a href="/A047600/b047600.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+2*x+x^2+x^3+3*x^4)/((1+x)*(1-x)^2*(1+x^2)).

%F a(n) = 2*n-1 -(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) = A047621(k), a(2k-1) = A047461(k). (End)

%F E.g.f.: (6 + sin(x) - 2*cos(x) + (4*x - 3)*sinh(x) + 4*(x - 1)*cosh(x))/2. - _Ilya Gutkovskiy_, Jun 03 2016

%F Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)+1)*Pi/16 - (4+3*sqrt(2))*log(2)/16 + 3*sqrt(2)*log(sqrt(2)+2)/8. - _Amiram Eldar_, Dec 24 2021

%p A047600:=n->2*n-1-(3+I^(2*n))*(1+I^(n*(n+1)))/4: seq(A047600(n), n=1..100); # _Wesley Ivan Hurt_, Jun 02 2016

%t Select[Range[120], MemberQ[{1,3,4,5}, Mod[#,8]]&] (* _Harvey P. Dale_, Mar 09 2011 *)

%t LinearRecurrence[{1, 0, 0, 1, -1}, {1, 3, 4, 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 [1,3,4,5]];

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

%o (PARI) Vec((1+2*x+x^2+x^3+3*x^4)/((1+x)*(1-x)^2*(1+x^2))+O(x^60)) (End)

%Y Cf. A047461, A047621.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_