A065176 Site swap sequence associated with the permutation A065174 of Z.

0, 2, 2, 4, 4, 2, 2, 8, 8, 2, 2, 4, 4, 2, 2, 16, 16, 2, 2, 4, 4, 2, 2, 8, 8, 2, 2, 4, 4, 2, 2, 32, 32, 2, 2, 4, 4, 2, 2, 8, 8, 2, 2, 4, 4, 2, 2, 16, 16, 2, 2, 4, 4, 2, 2, 8, 8, 2, 2, 4, 4, 2, 2, 64, 64, 2, 2, 4, 4, 2, 2, 8, 8, 2, 2, 4, 4, 2, 2, 16, 16, 2, 2, 4, 4, 2, 2, 8, 8, 2, 2, 4, 4, 2, 2, 32, 32, 2, 2

Offset: 1

Author

Antti Karttunen, Oct 19 2001

Keywords

• easy
• nonn

Comments

Here the site swap pattern ...,2,16,2,4,2,8,2,4,2,0,2,4,2,8,2,4,2,16,2,... 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, etc. successively.

This pattern is shown in the figure 7 of Buhler and Graham paper and uses infinitely many balls, with each ball at step t thrown always to constant "height" 2^A001511[abs(t)] (no balls in hands at step t=0).

Links

• Georg Fischer, Table of n, a(n) for n = 1..16384
• Joe Buhler and R. L. Graham, Juggling Drops and Descents, Amer. Math. Monthly, Vol. 101, No. 6 (1994), 507-519.

Crossrefs

Bisection of this gives A171977 or 2*A006519 or 2^A001511.

Cf. A065174.

Programs

Maple

[seq(TZ2(abs(N2Z(n))), n=1..120)];  # using TZ2 from A065174
N2Z := n -> ((-1)^n)*floor(n/2); Z2N := z -> 2*abs(z)+`if`((z < 1),1,0);
# Alternative:
A065176 := n -> `if`(n = 1, 0, 2^padic:-ordp(n - 1 + irem(n-1, 2), 2)):
seq(A065176(n), n = 1..99);  # Peter Luschny, Nov 14 2021

Mathematica

a[n_] := 2^IntegerExponent[n - Mod[n, 2], 2]; a[1] = 0; Array[a, 100] (* Amiram Eldar, May 22 2025 *)

PARI

a(n) = if(n==1,0, 1<Kevin Ryde, Jul 09 2021

Python

def A065176(n):
    s, h = 1, n // 2
    if 0 == h: return 0
    while 0 == h % 2:
        h //= 2
        s += s
    return s + s
print([A065176(n) for n in range(1, 100)])  # Peter Luschny, Nov 14 2021

Formula

G.f.: (1-x+x^2)/(1-x) + (1+x)*Sum(k>=1, 2^(k-1)*x^2^k/(1-x^2^k)). - Ralf Stephan, Apr 17 2003

a(n) = A171977(floor(n/2)) for n >= 2. - Georg Fischer, Nov 28 2022