A110962 Fractalization of A025480, zero-based version of Kimberling's paraphrase of the binary number system.

0, 0, 0, 0, 1, 0, 0, 0, 2, 1, 1, 0, 3, 0, 0, 0, 4, 2, 2, 1, 5, 1, 1, 0, 6, 3, 3, 0, 7, 0, 0, 0, 8, 4, 4, 2, 9, 2, 2, 1, 10, 5, 5, 1, 11, 1, 1, 0, 12, 6, 6, 3, 13, 3, 3, 0, 14, 7, 7, 0, 15, 0, 0, 0, 16, 8, 8, 4, 17, 4, 4, 2, 18, 9, 9, 2, 19, 2, 2, 1, 20, 10, 10, 5, 21, 5, 5, 1, 22, 11, 11, 1, 23, 1, 1, 0, 24, 12, 12

Offset

0,9

Comments

Self-descriptive sequence: the terms at odd indices are the sequence itself, while the terms at even indices (the skeleton of this sequence) are the terms of A025480, which is a zero-based version of Kimberling's paraphrase sequence, A003602.

Links

Antti Karttunen, Table of n, a(n) for n = 0..65537

Clark Kimberling, Fractal sequences

Index entries for sequences related to binary expansion of n

Formula

For even n, a(n) = A025480(n/2), for odd n, a(n) = a((n-1)/2). - Antti Karttunen, Apr 18 2022

a(2n+1) = a(4n+3) = a(n).

a(2n) = a(4n+1) = a(4n+2) = A025480(n/2).

a(4n) = a(8n+1) = a(8n+2) = n.

a(n) = A110963(1+n) - 1.

Mathematica

A110962[n_] := Nest[(BitShiftRight[#, IntegerExponent[#, 2]] + 1)/2 &, n + 1, 2] - 1;
Array[A110962, 100, 0] (* Paolo Xausa, Sep 12 2025 *)

Prog

(PARI)
A025480(n) = (n>>valuation(n*2+2, 2));
A110962(n) = if(!(n%2), A025480(n/2), A110962((n-1)/2)); \\ Antti Karttunen, Apr 18 2022
(PARI) a(n) = n++; n>>=valuation(n, 2); n>>valuation(2*n+2, 2); \\ Ruud H.G. van Tol, Jun 23 2024

Crossrefs

Cf. A000265, A003602, A025480, A089309, A103391, A110812, A110779, A110766.

One less than A110963 (note also the different starting offsets).

Sequence in context: A301636 A238857 A253587 * A065715 A180984 A051628

Adjacent sequences: A110959 A110960 A110961 * A110963 A110964 A110965

Keyword

base,easy,nonn

Author

Alexandre Wajnberg, Sep 26 2005

Extensions

Entry edited and more terms added by Antti Karttunen, Apr 18 2022

Edited to match Kimberling's terminology by Peter Munn, Sep 11 2025

Status

approved