A065173 Site swap sequence that rises infinitely after t=0. The associated delta sequence p(t)-t for the permutation of Z: A065171.

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, 15, 30, 16, 8, 17, 34, 18, 9, 19, 38, 20, 10, 21, 42, 22, 11, 23, 46, 24, 12, 25, 50, 26, 13, 27, 54, 28, 14, 29, 58, 30, 15, 31, 62, 32, 16, 33, 66, 34, 17, 35, 70, 36, 18, 37, 74, 38

Offset

1,3

Comments

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.

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Joe Buhler and R. L. Graham, Juggling Drops and Descents, Amer. Math. Monthly, 101, (no. 6) 1994, 507 - 519.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Formula

a(2*k+2) = k+1, a(4*k+1) = k, a(4*k+3) = 4*k+2. - Ralf Stephan, Jun 10 2005

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

a(n) = 2*a(n-4)-a(n-8) for n>8. - Colin Barker, Oct 29 2016

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

Example

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

Maple

[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);

Prog

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

Crossrefs

The other bisection gives A000027.

Sequence in context: A318607 A340106 A259824 * A330965 A098474 A153199

Adjacent sequences: A065170 A065171 A065172 * A065174 A065175 A065176

Keyword

nonn,easy

Author

Antti Karttunen, Oct 19 2001

Status

approved