A353366 -id:A353366 - cp's OEIS frontend

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

Alexandre Wajnberg, Sep 26 2005

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.

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

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).

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

a(n) = n>>=valuation(n,2); 1+n>>valuation(2*n+2,2); \\ Ruud H.G. van Tol, Jun 23 2024

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

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

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

A353367 Sum of A110963 and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 1, 0, 2, 0, 1, 1, 4, 0, 1, 0, 2, 4, 1, 0, 5, 0, 2, 2, 4, 0, 1, 4, 8, 5, 1, 0, -2, 0, 1, 4, 10, 4, 3, 0, 6, 8, 2, 0, 10, 0, 2, 8, 4, 0, 1, 1, 10, 10, 4, 0, 3, 8, 1, 6, 16, 0, 1, 0, 2, 15, 1, 16, 14, 0, 5, 4, 6, 0, 3, 0, 20, 6, 3, 4, -2, 0, 2, 9, 22, 0, 6, 20, 12, 16, 2, 0, 16, 8, 2, 2, 4, 12, 1, 0, 25, 24

Offset: 1

Antti Karttunen, Apr 18 2022

sign

Note the negative terms, in contrast to A349135, which apparently has none.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A003602, A110963, A353366.

Cf. also A349135, A353369.

up_to = 65537;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003602(n) = (1+(n>>valuation(n,2)))/2;
A110963(n) = if(n%2, A003602((1+n)/2), A110963(n/2));
v353366 = DirInverseCorrect(vector(up_to,n,A110963(n)));
A353366(n) = v353366[n];
A353367(n) = (A110963(n)+A353366(n));

a(n) = A110963(n) + A353366(n).
For n > 1, a(n) = -Sum_{d|n, 1A110963(d) * A353366(n/d).
For all n >= 1, a(4*n) = A110963(n), and a(8*n-4) = A003602(n).

A353368 Dirichlet inverse of A103391, "even fractal sequence".

Original entry on oeis.org

1, -2, -2, 1, -2, 4, -3, -1, 2, 2, -4, -3, -3, 4, 3, 0, -2, -10, -6, 1, 8, 4, -7, 3, 1, -2, -8, -1, -5, -4, -9, -1, 14, -10, 2, 17, -6, 4, 1, -1, -4, -22, -12, 1, -3, 4, -13, -1, 6, -14, -6, 11, -8, 28, 1, 1, 19, -10, -16, 3, -9, 4, -25, -1, 10, -42, -18, 25, 18, 0, -19, -17, -6, -14, -12, 5, 13, 12, -21, 3, 24

Offset: 1

Antti Karttunen, Apr 18 2022

sign

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A003602, A103391, A353369.

Cf. also A349134, A353366.

up_to = 65537;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003602(n) = (n/2^valuation(n, 2)+1)/2; \\ From A003602
A103391(n) = if(1==n,1,(1+A003602(n-1)));
v353368 = DirInverseCorrect(vector(up_to,n,A103391(n)));
A353368(n) = v353368[n];

a(1) = 1; a(n) = -Sum_{d|n, d < n} A103391(n/d) * a(d).

a(n) = A353369(n) - A103391(n).

cp's OEIS Frontend

A110963 Fractalization of Kimberling's paraphrases sequence beginning with 1.

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A353367 Sum of A110963 and its Dirichlet inverse.

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A353368 Dirichlet inverse of A103391, "even fractal sequence".

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