A131730 - OEIS

%I #21 Dec 11 2023 01:45:02

%S 0,-1,1,0,1,-2,2,-1,2,-3,3,-2,3,-4,4,-3,4,-5,5,-4,5,-6,6,-5,6,-7,7,-6,

%T 7,-8,8,-7,8,-9,9,-8,9,-10,10,-9,10,-11,11,-10,11,-12,12,-11,12,-13,

%U 13,-12,13,-14,14,-13,14,-15,15,-14,15,-16,16,-15,16,-17,17,-16,17,-18,18,-17,18,-19,19,-18,19,-20,20,-19,20,-21,21,-20

%N a(4n) = n, a(4n+1) = -n-1, a(4n+2) = n+1, a(4n+3) = -n.

%H G. C. Greubel, <a href="/A131730/b131730.txt">Table of n, a(n) for n = 0..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 _Wesley Ivan Hurt_, May 29 2016: (Start)

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

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

%F a(n) = (1+i^(2*n)-(1+3*i)*i^(-n)-(1-3*i)*i^n+2*n*i^(2n))/8 where i=sqrt(-1).

%F a(2k) = A110654(k), a(2k+1) = - A028242(k).

%F abs(a(-n-2)) = A246552(n). (End)

%F E.g.f.: (-3*sin(x) - cos(x) + x*sinh(x) - (x - 1)*cosh(x))/4. - _Ilya Gutkovskiy_, May 29 2016

%F a(n) = cos(n*Pi)*(2*n+1-2*cos(n*Pi/2)+cos(n*Pi)+6*sin(n*Pi/2))/8. - _Wesley Ivan Hurt_, Oct 01 2017

%F a(n) = (-1)^n*(n - 2*floor((n+1)/4) - floor((n+2)/4)). - _Ridouane Oudra_, Dec 10 2023

%p A131730:=n->(1+I^(2*n)-(1+3*I)*I^(-n)-(1-3*I)*I^n+2*n*I^(2*n))/8: seq(A131730(n), n=0..100); # _Wesley Ivan Hurt_, May 29 2016

%t Table[(1+I^(2n)-(1+3*I)*I^(-n)-(1-3*I)*I^n+2*n*I^(2 n))/8, {n, 0, 80}] (* _Wesley Ivan Hurt_, May 29 2016 *)

%t LinearRecurrence[{-1, 0, 0, 1, 1}, {0, -1, 1, 0, 1}, 50] (* _G. C. Greubel_, May 29 2016 *)

%Y Cf. A028242, A110654, A246552.

%K sign,easy

%O 0,6

%A _Paul Curtz_, Sep 17 2007