A065173 - OEIS

%I #19 May 02 2017 22:17:15

%S 0,1,2,2,1,3,6,4,2,5,10,6,3,7,14,8,4,9,18,10,5,11,22,12,6,13,26,14,7,

%T 15,30,16,8,17,34,18,9,19,38,20,10,21,42,22,11,23,46,24,12,25,50,26,

%U 13,27,54,28,14,29,58,30,15,31,62,32,16,33,66,34,17,35,70,36,18,37,74,38

%N Site swap sequence that rises infinitely after t=0. The associated delta sequence p(t)-t for the permutation of Z: A065171.

%C Here the site swap pattern ..., 5, 18, 4, 14, 3, 10, 2, 6, 1, 2, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... that spans over the Z (zero throw is at t=0) has been folded to N by picking values at t=0, t=1, t=-1, t=2, t=-2, t=3, t=-3, etc. successively.

%H Colin Barker, <a href="/A065173/b065173.txt">Table of n, a(n) for n = 1..1000</a>

%H Joe Buhler and R. L. Graham, <a href="http://www.cecm.sfu.ca/organics/papers/buhler/index.html">Juggling Drops and Descents</a>, Amer. Math. Monthly, 101, (no. 6) 1994, 507 - 519.

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,2,0,0,0,-1).

%F a(2*k+2) = k+1, a(4*k+1) = k, a(4*k+3) = 4*k+2. - _Ralf Stephan_, Jun 10 2005

%F G.f.: x^2*(2*x^5+x^4+x^3+2*x^2+2*x+1) / ((x-1)^2*(x+1)^2*(x^2+1)^2). - _Colin Barker_, Feb 18 2013

%F a(n) = 2*a(n-4)-a(n-8) for n>8. - _Colin Barker_, Oct 29 2016

%F a(n) = (9*n-5-(n-5)*(-1)^n-3*(n-1)*(1-(-1)^n)*(-1)^((2*n-1+(-1)^n)/4))/16. - _Luce ETIENNE_, Oct 29 2016

%e G.f. = x^2 + 2*x^3 + 2*x^4 + x^5 + 3*x^6 + 6*x^7 + 4*x^8 + 2*x^9 + ...

%p [seq((InfRisingSS(N2Z(n))-N2Z(n)), n=1..120)]; N2Z := n -> ((-1)^n)*floor(n/2); Z2N := z -> 2*abs(z)+`if`((z < 1),1,0);

%o (PARI) concat(0, Vec(x^2*(2*x^5+x^4+x^3+2*x^2+2*x+1)/((x-1)^2*(x+1)^2*(x^2+1)^2) + O(x^100))) \\ _Colin Barker_, Oct 29 2016

%o (PARI) {a(n) = if( n%2==0, n/2, n%4==1, n\4, n-1)}; /* _Michael Somos_, Nov 06 2016 */

%Y The other bisection gives A000027.

%K nonn,easy

%O 1,3

%A _Antti Karttunen_, Oct 19 2001