OFFSET
1,4
COMMENTS
The minimum number k(n) of iterations in order to have u(k(n)) = a(n) is asymptotic to log(n)/2. Let m be any fixed positive integer and let Fr(m,n) = 3*Sum_{k = 1..n} a(k) - n^2 + m*n; then Fr(m,n) is a fractal generator function, i.e., there is an integer B(m) such that the graph for Fr(n,m) presents same fractal aspects for 1 <= n <= B(m). B(m) depends on the parity of m. B(2*p+1) = (5/3)*(4^p-1); B(2*p) = (2/3)*(4^p-1). [Formula for Fr(m,n) corrected by Petros Hadjicostas, Oct 21 2019 using the PARI program below.]
LINKS
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(n) is asymptotic to 2*n/3.
PROG
(PARI) 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); psplothraw(indices, fractal, 1); \\ To generate graphs
CROSSREFS
KEYWORD
nonn
AUTHOR
Benoit Cloitre and Boris Gourevitch (boris(AT)pi314.net), Aug 16 2002
STATUS
approved