A047605 - OEIS

%I #27 Sep 08 2022 08:44:57

%S 0,2,3,5,8,10,11,13,16,18,19,21,24,26,27,29,32,34,35,37,40,42,43,45,

%T 48,50,51,53,56,58,59,61,64,66,67,69,72,74,75,77,80,82,83,85,88,90,91,

%U 93,96,98,99,101,104,106,107,109,112,114,115,117,120,122,123

%N Numbers that are congruent to {0, 2, 3, 5} 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 _Bruno Berselli_, Dec 05 2011: (Start)

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

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

%F From _Wesley Ivan Hurt_, Jun 04 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(2k) = A047617(k), a(2k-1) = A047470(k). (End)

%F E.g.f.: (6 + sin(x) - cos(x) + (4*x - 5)*exp(x))/2. - _Ilya Gutkovskiy_, Jun 05 2016

%F Sum_{n>=2} (-1)^n/a(n) = (3-2*sqrt(2))*Pi/16 + 3*log(2)/8. - _Amiram Eldar_, Dec 21 2021

%p A047605:=n->2*(n-1)-(I^(n*(n+1))+1)/2: seq(A047605(n), n=1..100); # _Wesley Ivan Hurt_, Jun 04 2016

%t Table[(1+I)*((4-4*I)*n+5*I-5+I^(1-n)-I^n)/4, {n, 80}] (* _Wesley Ivan Hurt_, Jun 04 2016 *)

%t Flatten[#+{0,2,3,5}&/@(8*Range[0,20])] (* or *) LinearRecurrence[{2,-2,2,-1},{0,2,3,5},100] (* _Harvey P. Dale_, Sep 30 2018 *)

%o (PARI) a(n)=n\4*8+[-3,0,2,3][n%4+1] \\ _Charles R Greathouse IV_, Dec 05 2011

%o (Magma) [n : n in [0..150] | n mod 8 in [0, 2, 3, 5]]; // _Wesley Ivan Hurt_, Jun 04 2016

%Y Cf. A047470, A047617.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_