

A061420


a(n) = a(ceiling((n1)*2/3)) + 1 with a(0) = 0.


2



0, 1, 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10
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OFFSET

0,3


COMMENTS

Least k such that f^(k)(n) = 0 where f(x) = floor(2/3*x) and f^(k+1)(x) = f(f^(k)(x)).  Benoit Cloitre, May 26 2007


LINKS

Clark Kimberling, Table of n, a(n) for n = 0..2000
William J. Gilbert, Radix Representations of Quadratic Fields, Journal of Mathematical Analysis and Applications 83 (1981) pp 264274. Gilbert (page 273) cites Wang and Washburn (below) in connection with the length of the base 3/2 expansion of the even positive integers.
A. M. Odlyzko and H. S. Wilf, Functional iteration and the Josephus problem, Glasgow Math. J. 33, 235240, 1991.
E. T. H. Wang, Phillip C. Washburn, Problem E2604, American Mathematical Monthly 84 (1977) pp. 821822.


FORMULA

a(n) = a(n1) + 1 if n is in A061419; a(n) = a(n1) otherwise.
From Clark Kimberling, Oct 19 2012: (Start)
a(n) = a(floor(2*n/3)) + 1, where a(0) = 0 (alternative definition).
Washburn's solution of Problem E2604 (see References) shows that (for n>0), a(n) = floor(L((n+1)/c)), where L is the logarithm with base 3/2 and
c = lim_{n>infinity} (2/3)^n*s(n) where s(n) = floor(3*s(n1)/2) + 1 and s(0)=0. The editors state that "It may be interesting to know whether c is irrational or even transcendental"; c = 1.62227050288476731595695098289932... .
Odlyzko and Wilf also discuss the defining recurrence, and they, after Washburn, give a formula for the sequence using c, as in the third Mathematica program below.
(End)


EXAMPLE

a(10) = a(ceiling(9*2/3)) + 1 = a(6) + 1 = 4 + 1 = 5.


MAPLE

a:= n> `if`(n=0, 0, a(ceil((n1)*2/3))+1):
seq(a(n), n=0..100); # Alois P. Heinz, Oct 29 2012


MATHEMATICA

(* 1st program, using the alternative definition *)
a[0] = 0; a[n_] := a[Floor[2 n/3]] + 1;
Table[a[n], {n, 0, 120}]
(* 2nd program, using Cloitre's recurrence *)
f[x_] := Floor[2 x/3]; g[0, x_] := f[x];
g[k_, x_] := f[g[k  1, x]];
u[n_] := Flatten[Table[g[k, n], {k, 0, 12}]]
v[n_] := First[Position[u[n], 0]];
Flatten[Table[v[n], {n, 1, 120}]]
(* 3rd program, using the constant c *)
f[n_] := Floor[Log[3/2, (n + 1)/1.62227050288476731595695098289932]]
Table[f[n], {n, 1, 120}]
(* Clark Kimberling, Oct 23 2012 *)


PROG

(PARI) a(n) = if(n<0, 0, s=n; c=0; while(floor(s)>0, s=floor(2/3*s); c++); c) \\ Benoit Cloitre, May 26 2007
(MAGMA) [IsZero(n) select 0 else Self(Floor(2*n/3)+1)+1: n in [0..90]]; // Bruno Berselli, Oct 31 2012


CROSSREFS

Cf. A029837, A061419, A083286 (the constant c).
Sequence in context: A083398 A221671 A301640 * A003057 A239308 A216256
Adjacent sequences: A061417 A061418 A061419 * A061421 A061422 A061423


KEYWORD

nonn


AUTHOR

Henry Bottomley, May 02 2001


STATUS

approved



