

A071162


Simple rewriting of binary expansion of n resulting A014486codes for rooted binary trees with height equal to number of internal vertices. (Binary trees where at each internal vertex at least the other child is leaf).


7



0, 2, 10, 12, 42, 44, 52, 56, 170, 172, 180, 184, 212, 216, 232, 240, 682, 684, 692, 696, 724, 728, 744, 752, 852, 856, 872, 880, 936, 944, 976, 992, 2730, 2732, 2740, 2744, 2772, 2776, 2792, 2800, 2900, 2904, 2920, 2928, 2984, 2992, 3024, 3040, 3412, 3416
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OFFSET

0,2


COMMENTS

Essentially rewrites in binary expansion of n each 0 > 01, 1X > 1(rewrite X)0, where X is the maximal suffix after the 1bit, which will be rewritten recursively (see the given Schemefunction). Because of this, the terms of the binary length 2n are counted by 2's powers, A000079.
In rooted plane (general) tree context, these are those totally balanced binary sequences (terms of A014486) where nonleaf subtrees can occur only as the rightmost branch (at any level of a general tree), but nowhere else. (Cf. A209642).
Also, these are exactly those rooted plane trees whose Łukasiewicz words happen to be valid asynchronous siteswap juggling patterns. (This was the original, albeit quite frivolous definition of this sequence for almost ten years 20022012. Cf. A071160.)


LINKS

Antti Karttunen, Table of n, a(n) for n = 0..65535
OEIS Wiki, Łukasiewicz words
Index entries for sequences related to Łukasiewicz


PROG

(Scheme): (define (A071162 n) (let loop ((n n) (s 0) (i 1)) (cond ((zero? n) s) ((even? n) (loop (/ n 2) (+ s i) (* i 4))) (else (loop (/ ( n 1) 2) (* 2 (+ s i)) (* i 4))))))
(Python)
def a036044(n): return int(''.join('1' if i == '0' else '0' for i in bin(n)[2:][::1]), 2)
def a209642(n):
s=0
i=1
while n!=0:
if n%2==0:
n//=2
s=4*s + 1
else:
n=(n  1)//2
s=(s + i)*2
i*=4
return s
def a(n): return 0 if n==0 else a036044(a209642(n))
print([a(n) for n in range(101)]) # Indranil Ghosh, May 25 2017


CROSSREFS

a(n) = A014486(A071163(n)) = A036044(A209642(n)) = A056539(A209642(n)).
A209859 provides an "inverse" function, i.e. A209859(a(n)) = n for all n.
Cf. A209636, A209637, A209862.
Sequence in context: A014486 A166751 A216649 * A075165 A209641 A209642
Adjacent sequences: A071159 A071160 A071161 * A071163 A071164 A071165


KEYWORD

nonn,base


AUTHOR

Antti Karttunen, May 14 2002


STATUS

approved



