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Numbers that are congruent to {0, 3, 6, 7} mod 8.
4

%I #36 Dec 23 2021 10:04:55

%S 0,3,6,7,8,11,14,15,16,19,22,23,24,27,30,31,32,35,38,39,40,43,46,47,

%T 48,51,54,55,56,59,62,63,64,67,70,71,72,75,78,79,80,83,86,87,88,91,94,

%U 95,96,99,102,103,104,107,110,111,112,115,118,119,120,123,126

%N Numbers that are congruent to {0, 3, 6, 7} mod 8.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-1).

%F From _R. J. Mathar_, Oct 08 2011: (Start)

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

%F a(n) = 2*n-1-sin(Pi*n/2). (End)

%F a(n) = 2n-1-cos(Pi*(n-1)/2). - _Wesley Ivan Hurt_, Oct 22 2013

%F From _Wesley Ivan Hurt_, May 20 2016: (Start)

%F a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-a(n-4) for n>4.

%F a(n) = 2*n-2-I^(1-n)*(I^(n-1)-1)^2/2 where I=sqrt(-1).

%F a(2n) = A004767(n-1) for n>0, a(2n-1) = A047451(n). (End)

%F Sum_{n>=2} (-1)^n/a(n) = 5*log(2)/8 - Pi/16. - _Amiram Eldar_, Dec 23 2021

%p A047557:=n->2*n-1-cos(Pi*(n-1)/2): seq(A047557(k), k=1..100); # _Wesley Ivan Hurt_, Oct 22 2013

%t Table[2n-1-Cos[Pi(n-1)/2], {n,100}] (* _Wesley Ivan Hurt_, Oct 22 2013 *)

%o (Sage) [lucas_number1(n,0,1)+2*n+3 for n in range(-1,55)] # _Zerinvary Lajos_, Jul 06 2008

%Y Cf. A004767, A047404, A047431, A047451, A047546, A047578, A056594.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_

%E More terms from _Wesley Ivan Hurt_, May 20 2016