A073504 A possible basis for finite fractal sequences: let u(1) = 1, u(2) = n, u(k) = floor(u(k-1)/2) + floor(u(k-2)/2); then a(n) = lim_{k->infinity} u(k).

0, 0, 0, 2, 2, 2, 2, 4, 4, 4, 4, 6, 6, 8, 8, 10, 10, 10, 10, 12, 12, 12, 12, 14, 14, 14, 14, 16, 16, 18, 18, 20, 20, 20, 20, 22, 22, 22, 22, 24, 24, 24, 24, 26, 26, 28, 28, 30, 30, 30, 30, 32, 32, 34, 34, 36, 36, 36, 36, 38, 38, 40, 40, 42, 42, 42, 42, 44, 44, 44, 44, 46, 46, 46

Offset: 1

Author

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

Keywords

• nonn

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

• 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).

Crossrefs

Cf. A073059 and A071992 (curiously A071992 presents the same fractal aspects as Fr(n, m)).

Programs

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); myStr=plothrawexport("svg",indices,fractal,1);write("myPlot.svg",myStr); \\ To generate graphs

Formula

a(n) is asymptotic to 2*n/3.