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%I #29 Jan 04 2024 11:36:39
%S 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,
%T 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,
%U 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
%N Fractalization of Kimberling's paraphrases sequence beginning with 1.
%C 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.
%H Antti Karttunen, <a href="/A110963/b110963.txt">Table of n, a(n) for n = 1..65537</a>
%H Clark Kimberling, <a href="http://faculty.evansville.edu/ck6/integer/fractals.html">Fractal sequences</a>.
%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F For even n, a(n) = a(n/2), for odd n, a(n) = A003602((1+n)/2). - _Antti Karttunen_, Apr 03 2022
%F For n >= 0, (Start)
%F a(4n+2) = a(4n+3) = A003602(1+n).
%F a(8n+1) = A005408(n) = 2*n + 1.
%F a(4n+1) = a(8n+2) = a(8n+3) = 1+n.
%F a(n) = A110962(n-1) + 1.
%F (End)
%F a(n) = A353367(4*n). - _Antti Karttunen_, Apr 20 2022
%o (PARI)
%o A003602(n) = (1+(n>>valuation(n,2)))/2;
%o A110963(n) = if(n%2, A003602((1+n)/2), A110963(n/2)); \\ _Antti Karttunen_, Apr 03 2022
%o (Python)
%o 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
%Y One more than A110962 (but note the different starting offsets).
%Y Cf. A000265, A003602, A005408, A110812, A110779, A110766, A351565 [= 2*a(n) - 1].
%Y Cf. A353366 (Dirichlet inverse), A353367 (sum with it).
%K base,easy,nonn
%O 1,5
%A _Alexandre Wajnberg_, Sep 26 2005
%E 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
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