OFFSET
1,1
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
B. Cloitre, Graph of Fr(n,4) for 1 <= n <= B(4)
B. Cloitre, Graph of Fr(n,6) for 1 <= n <= B(6)
B. Cloitre, Graph of Fr(n,8) for 1 <= n <= B(8)
B. Cloitre, Graph of Fr(n,5) for 1 <= n <= B(5)
B. Cloitre, Graph of Fr(n,7) for 1 <= n <= B(7)
B. Cloitre, Graph of Fr(n,9) for 1 <= n <= B(9)
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
PROG
(PARI) \\ To generate graphs:
for(n = 1, taille, u1=1; u2=n; while((u2!=u1)||((u2%2)==1), u3=u2; u2=floor(u2/2)+fl oor(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); psplothraw(indices, fractal, 1);
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Benoit Cloitre and Boris Gourevitch (boris(AT)pi314.net), Aug 16 2002
EXTENSIONS
Erroneous formula removed by Antti Karttunen, Oct 09 2018
STATUS
approved