A110963 Fractalization of Kimberling's paraphrase of the binary number system beginning with 1.

1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 2, 1, 4, 1, 1, 1, 5, 3, 3, 2, 6, 2, 2, 1, 7, 4, 4, 1, 8, 1, 1, 1, 9, 5, 5, 3, 10, 3, 3, 2, 11, 6, 6, 2, 12, 2, 2, 1, 13, 7, 7, 4, 14, 4, 4, 1, 15, 8, 8, 1, 16, 1, 1, 1, 17, 9, 9, 5, 18, 5, 5, 3, 19, 10, 10, 3, 20, 3, 3, 2, 21, 11, 11, 6, 22, 6, 6, 2, 23, 12, 12, 2, 24, 2, 2, 1, 25, 13

Offset

1,5

Comments

Self-descriptive sequence: terms at even indices are the sequence itself, terms at odd indices (the skeleton of this sequence) are the terms of Kimberling's paraphrase sequence, A003602.

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Clark Kimberling, Fractal sequences.

Index entries for sequences related to binary expansion of n

Formula

For even n, a(n) = a(n/2), for odd n, a(n) = A003602((1+n)/2). - Antti Karttunen, Apr 03 2022

For n >= 0, (Start)

a(4n+2) = a(4n+3) = A003602(1+n).

a(8n+1) = A005408(n) = 2*n + 1.

a(4n+1) = a(8n+2) = a(8n+3) = 1+n.

a(n) = A110962(n-1) + 1.

(End)

a(n) = A353367(4*n). - Antti Karttunen, Apr 20 2022

a(n) = A003602(A003602(n)). - Ridouane Oudra, Dec 28 2024

Mathematica

A110963[n_] := Nest[(BitShiftRight[#, IntegerExponent[#, 2]] + 1)/2 &, n, 2];
Array[A110963, 100] (* Paolo Xausa, Sep 12 2025 *)

Prog

(PARI)
A003602(n) = (1+(n>>valuation(n, 2)))/2;
A110963(n) = if(n%2, A003602((1+n)/2), A110963(n/2)); \\ Antti Karttunen, Apr 03 2022
(PARI) a(n) = n>>=valuation(n, 2); 1+n>>valuation(2*n+2, 2); \\ Ruud H.G. van Tol, Jun 23 2024
(Python)
def A110963(n): return (1+(m:=n>>(~n&n-1).bit_length())>>(m+1&-m-1).bit_length())+1 # Chai Wah Wu, Jan 04 2024

Crossrefs

One more than A110962 (but note the different starting offsets).

Cf. A000265, A003602, A005408, A089309, A110812, A110779, A110766, A351565 [= 2*a(n) - 1].

Cf. A353366 (Dirichlet inverse), A353367 (sum with it).

Sequence in context: A288003 A304382 A304717 * A292622 A292869 A106348

Adjacent sequences: A110960 A110961 A110962 * A110964 A110965 A110966

Keyword

base,easy,nonn

Author

Alexandre Wajnberg, Sep 26 2005

Extensions

Entry edited, starting offset corrected (from 0 to 1), and the offsets in formulas changed accordingly, and more terms added by Antti Karttunen, Apr 03 2022

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

Status

approved