OFFSET
1,2
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, 101, (no. 6) 1994, 507 - 519.
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
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
PROG
(PARI) a(n) = if(n==1, 0, 1<<valuation(bitnegimply(n, 1), 2)); \\ 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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Antti Karttunen, Oct 19 2001
STATUS
approved