A065171 Permutation of Z, folded to N, corresponding to the site swap pattern ...26120123456... which ascends infinitely after t=0.

1, 4, 2, 8, 3, 12, 6, 16, 5, 20, 10, 24, 7, 28, 14, 32, 9, 36, 18, 40, 11, 44, 22, 48, 13, 52, 26, 56, 15, 60, 30, 64, 17, 68, 34, 72, 19, 76, 38, 80, 21, 84, 42, 88, 23, 92, 46, 96, 25, 100, 50, 104, 27, 108, 54, 112, 29, 116, 58, 120, 31, 124, 62, 128, 33, 132, 66, 136, 35

Offset: 1

Author

Antti Karttunen, Oct 19 2001

Keywords

• nonn
• easy

Comments

This permutation consists of one fixed point (at 0, mapped here to 1) and an infinite number of infinite cycles.

Examples

			G.f. = x + 4*x^2 + 2*x^3 + 8*x^4 + 3*x^5 + 12*x^6 + 6*x^7 + 16*x^8 + ...
		

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 sequences that are permutations of the natural numbers
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Crossrefs

Inverse permutation: A065172. A065173 gives the deltas p(t)-t, i.e., the associated site swap sequence. Cf. also A065167, A065174, A065260.

Programs

Maple

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

PARI

Vec(x*(2*x^6+4*x^5+x^4+8*x^3+2*x^2+4*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, n\2+1, n*2)}; /* Michael Somos, Nov 06 2016 */

Formula

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

G.f.: x*(2*x^6+4*x^5+x^4+8*x^3+2*x^2+4*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) = (11*n-1+(5*n+1)*(-1)^n+(n-3)*(1-(-1)^n)*(-1)^((2*n+3+(-1)^n)/4))/8. - Luce ETIENNE, Oct 20 2016