A162598 Ordinal transform of A265332.

1, 1, 2, 1, 3, 4, 2, 1, 5, 6, 7, 3, 8, 4, 2, 1, 9, 10, 11, 12, 5, 13, 14, 6, 15, 7, 3, 16, 8, 4, 2, 1, 17, 18, 19, 20, 21, 9, 22, 23, 24, 10, 25, 26, 11, 27, 12, 5, 28, 29, 13, 30, 14, 6, 31, 15, 7, 3, 32, 16, 8, 4, 2, 1, 33, 34, 35, 36, 37, 38, 17, 39, 40, 41, 42, 18, 43, 44, 45, 19, 46, 47

Offset: 1

Author

Franklin T. Adams-Watters, Jul 07 2009

Keywords

• nonn
• look

Comments

This is a fractal sequence.

It appears that each group of 2^k terms starts with 1 and ends with the remaining powers of two from 2^k down to 2^1.

From Antti Karttunen, Jan 09-12 2016: (Start)

This is ordinal transform of A265332, which is modified A051135 (with a(1) = 1, instead of 2). - after Franklin T. Adams-Watters' original definition for this sequence.

A000079 (powers of 2) indeed gives the positions of ones in this sequence. This follows from the properties (3) and (4) of A004001 given on page 227 of Kubo & Vakil paper (page 3 of PDF), which together also imply the pattern observed above, more clearly represented as:

a(2) = 1.

a(3..4) = 2, 1.

a(6..8) = 4, 2, 1.

a(13..16) = 8, 4, 2, 1.

a(28..31) = 16, 8, 4, 2, 1.

etc.

(End)

Links

• Antti Karttunen, Table of n, a(n) for n = 1..8192
• T. Kubo and R. Vakil, On Conway's recursive sequence, Discr. Math. 152 (1996), 225-252.
• Index entries for Hofstadter-type sequences

Crossrefs

Cf. A004001, A051135, A080677, A087686.

Row index of A265901, column index of A265903.

Cf. A265332 (corresponding other index).

Cf. A000079 (positions of ones).

Cf. A000225 (from the term 3 onward the positions of 2's).

Cf. A000325 (from its third term 5 onward the positions of 3's, which occur always as the last term before the next descending subsequence of powers of two).

Programs

Mathematica

terms = 100;
h[1] = 1; h[2] = 1;
h[n_] := h[n] = h[h[n - 1]] + h[n - h[n - 1]];
t = Array[h, 2*terms];
A051135 = Take[Transpose[Tally[t]][[2]], terms];
b[_] = 1;
a[n_] := a[n] = With[{t = If[n == 1, 1, A051135[[n]]]}, b[t]++];
Array[a, terms] (* Jean-François Alcover, Dec 19 2021, after Robert G. Wilson v in A051135 *)

Formula

Let b(1) = 1, b(n) = A051135(n) for n > 1. Then a(n) is the number of b(k) that equal b(n) for 1 <= k <= n: sum( 1, 1<=k<=n and a(k)=a(n) ).

If A265332(n) = 1, then a(n) = A004001(n), otherwise a(n) = a(A080677(n)-1) = a(n - A004001(n)). - Antti Karttunen, Jan 09 2016

Extensions

Name amended by Antti Karttunen, Jan 09 2016