

A110963


Fractalization of Kimberling's paraphrases sequence beginning with 1.


5



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,5


COMMENTS

Selfdescriptive 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


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(n1) + 1.
(End)
a(n) = A353367(4*n).  Antti Karttunen, Apr 20 2022


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


CROSSREFS

One more than A110962 (but note the different starting offsets).
Cf. A000265, A003602, A005408, 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


STATUS

approved



