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A073504
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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).
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2
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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
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OFFSET
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1,4
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COMMENTS
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The minimum number k(n) of iterations in order to have u(k(n)) = a(n) is asymptotic to ln(n)/2. Let m be any fixed positive integer and let Fr(m,n) = 3*Sum( k=1,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(2p+1)=(5/3)*(4^p-1); B(2p)=(2/3)*(4^p-1).
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LINKS
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Table of n, a(n) for n=1..74.
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)
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FORMULA
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a(n) is asymptotic to 2n/3 To generate graphs (PARI) 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);
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CROSSREFS
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Cf. A073059 & A071992 (curiously A071992 presents the same fractal aspects as Fr(n, m)).
Sequence in context: A122461 A092533 A092532 * A092508 A032544 A200675
Adjacent sequences: A073501 A073502 A073503 * A073505 A073506 A073507
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KEYWORD
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nonn
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AUTHOR
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Benoit Cloitre and Boris Gourevitch (boris(AT)pi314.net), Aug 16 2002
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STATUS
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approved
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