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Numbers that are congruent to {0, 1, 5, 7} mod 8.
1

%I #38 Sep 08 2022 08:44:57

%S 0,1,5,7,8,9,13,15,16,17,21,23,24,25,29,31,32,33,37,39,40,41,45,47,48,

%T 49,53,55,56,57,61,63,64,65,69,71,72,73,77,79,80,81,85,87,88,89,93,95,

%U 96,97,101,103,104,105,109,111,112,113,117,119,120,121,125

%N Numbers that are congruent to {0, 1, 5, 7} mod 8.

%H Vincenzo Librandi, <a href="/A047479/b047479.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 _Colin Barker_, May 14 2012: (Start)

%F a(n) = (-7-(-1)^n+(2-i)*(-i)^n+(2+i)*i^n+8*n)/4 where i=sqrt(-1).

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

%F a(n) = a(n-1) + a(n-4) - a(n-5) for n>5. - _Vincenzo Librandi_, May 16 2012

%F a(2k) = A047522(k), a(2k-1) = A047615(k). - _Wesley Ivan Hurt_, Jun 01 2016

%F E.g.f.: (2 - sin(x) + 2*cos(x) + (4*x - 3)*sinh(x) + 4*(x - 1)*cosh(x))/2. - _Ilya Gutkovskiy_, Jun 02 2016

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

%p A047479:=n->(-7-I^(2*n)+(2-I)*(-I)^n+(2+I)*I^n+8*n)/4: seq(A047479(n), n=1..100); # _Wesley Ivan Hurt_, Jun 01 2016

%t Select[Range[0,300], MemberQ[{0,1,5,7}, Mod[#,8]]&] (* _Vincenzo Librandi_, May 16 2012 *)

%o (Magma) I:=[0, 1, 5, 7, 8]; [n le 5 select I[n] else Self(n-1)+Self(n-4)-Self(n-5): n in [1..70]]; // _Vincenzo Librandi_, May 16 2012

%o (PARI) my(x='x+O('x^100)); concat(0, Vec(x^2*(1+4*x+2*x^2+x^3)/((1-x)^2*(1+x)*(1+x^2)))) \\ _Altug Alkan_, Dec 24 2015

%Y Cf. A047522, A047615.

%K nonn,easy

%O 1,3

%A _N. J. A. Sloane_