A073059 a(n) = (1/2)*(A073504(n+1) - A073504(n)).

0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0

Offset: 1

Author

Benoit Cloitre and Boris Gourevitch (boris(AT)pi314.net), Aug 16 2002

Keywords

• nonn

Comments

Let m be any fixed positive integer and let Fr(m,n) := 3*Sum_{k = 1..n} A073504(k) - n^2 + m*n. Then Fr(m,n) allows us to generate fractal sequences, i.e., there is an integer B(m) such that the graph for Fr(n,m) is fractal-like for 1 <= n <= B(m). B(m) depends on the parity of m: B(2*p+1) = (5/3)*(4^p - 1) and B(2*p) =(2/3)*(4^p - 1). [Formula for Fr(m,n) corrected by Petros Hadjicostas, Oct 21 2019]

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Benoit Cloitre, Graph of Fr(n,4) for 1 <= n <= B(4)
• Benoit Cloitre, Graph of Fr(n,6) for 1 <= n <= B(6)
• Benoit Cloitre, Graph of Fr(n,8) for 1 <= n <= B(8)
• Benoit Cloitre, Graph of Fr(n,5) for 1 <= n <= B(5)
• Benoit Cloitre, Graph of Fr(n,7) for 1 <= n <= B(7)
• Benoit Cloitre, Graph of Fr(n,9) for 1 <= n <= B(9)
• Index entries for characteristic functions

Crossrefs

Not the same as the period-doubling sequence A096268!

Cf. A073504 and A071992 (curiously A071992 presents the same fractal aspects as Fr(n, m)).

Cf. A098725.

Programs

PARI

\\ To generate graphs:
for(n = 1,taille,u1=1; u2=n; while((u2!=u1)||((u2%2)==1),u3=u2; u2=floor(u2/2)+floor(u1/2); u1=u3; ); b[n]=u2; ) fr(m,k)=(3*sum(i=1,k,b[i]))-k^2+m*k; bound(m)=if((m%2)==1,p=(m-1)/2; 5/3*(4^p-1),2/3*(4^(m/2)-1)); m=5; fractal=vector(bound(m)); for(i=1,bound(m),fractal[i]=fr(m,i); ); Mm=vecmax(fractal) indices=vector(bound(m)); for(i=1,bound(m), indices[i]=i); myStr=plothrawexport("svg",indices,fractal,1);write("myPlot.svg",myStr);

PARI

A073059(n) = if(1==n,0,if(!(n%2),0,if(3==(n%4),1,A073059((n-1)/4)))); \\ Antti Karttunen, Oct 09 2018, after Ralf Stephan's Dec 11 2004 formula

PARI

up_to = 10000;
A073504list(up_to) = { my(v=vector(up_to)); for(n=1, up_to, u1=1; u2=n; while((u2!=u1)||((u2%2) == 1), u3=u2; u2=(u2\2)+(u1\2); u1=u3); v[n]=u2); (v); };
v073504 = A073504list(up_to);
A073504(n) = v073504[n];
A073059(n) = (1/2)*(A073504(n+1)-A073504(n)); \\ Antti Karttunen, Nov 27 2018, after code sent by Benoit Cloitre (personal communication), implementing the original definition

Python

from functools import lru_cache
@lru_cache(maxsize=None)
def A073059(n): return (1 if n&2 else A073059(n>>2)) if n&1 else 0 # Chai Wah Wu, Feb 07 2025

Formula

a(4*k+3) = 1, a(4*k+2) = a(4*k+4) = 0, a(16*k+13) = 1, ...

A073504(n) = Sum_{k = 1..n} a(k) is asymptotic to 2*n/3.

a(2*n) = 0, a(4*n+3) = 1, a(4*n+1) = a(n). - Ralf Stephan, Dec 11 2004

Extensions

Erroneous formula removed by Antti Karttunen, Oct 09 2018