A110963 Fractalization of Kimberling's paraphrases sequence 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

Author

Alexandre Wajnberg, Sep 26 2005

Keywords

• base
• easy
• nonn

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 paraphrases sequence (A003602) beginning with 1.

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

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).

Programs

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

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

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