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%I #33 Oct 22 2019 04:20:26
%S 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,
%T 16,18,18,20,20,20,20,22,22,22,22,24,24,24,24,26,26,28,28,30,30,30,30,
%U 32,32,34,34,36,36,36,36,38,38,40,40,42,42,42,42,44,44,44,44,46,46,46
%N 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).
%C 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.]
%H B. Cloitre, <a href="/A073059/a073059_4.jpg">Graph of Fr(n,4) for 1 <= n <= B(4)</a>.
%H B. Cloitre, <a href="/A073059/a073059_6.jpg">Graph of Fr(n,6) for 1 <= n <= B(6)</a>.
%H B. Cloitre, <a href="/A073059/a073059_8.jpg">Graph of Fr(n,8) for 1 <= n <= B(8)</a>.
%H B. Cloitre, <a href="/A073059/a073059_5.jpg">Graph of Fr(n,5) for 1 <= n <= B(5)</a>.
%H B. Cloitre, <a href="/A073059/a073059_7.jpg">Graph of Fr(n,7) for 1 <= n <= B(7)</a>.
%H B. Cloitre, <a href="/A073059/a073059_9.jpg">Graph of Fr(n,9) for 1 <= n <= B(9)</a>.
%F a(n) is asymptotic to 2*n/3.
%o (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
%Y Cf. A073059 and A071992 (curiously A071992 presents the same fractal aspects as Fr(n, m)).
%K nonn
%O 1,4
%A _Benoit Cloitre_ and Boris Gourevitch (boris(AT)pi314.net), Aug 16 2002