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

1,2

Comments

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

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).

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

Example

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

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);

Prog

(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 */

Crossrefs

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

Sequence in context: A026209 A198473 A133640 * A026192 A026142 A095399

Adjacent sequences: A065168 A065169 A065170 * A065172 A065173 A065174

Keyword

nonn,easy

Author

Antti Karttunen, Oct 19 2001

Status

approved