1,1

Let m be any fixed positive integer and let Fr(m,n) := 3*Sum( k = 1, 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(2p+1)=(5/3)*(4^p-1) and B(2p)=(2/3)*(4^p-1).

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)

a(4k+3)= 1, a(4k+2)=a(4k+4)=0, a(16k+13) = 1 ... A073504 (n)=sum(k = 1, n, a(k)) is asymptotic to 2n/3.

For n>1, a(n) = 1-A035263(n-1).

a(2n) = 0, a(4n+3) = 1, a(4n+1) = a(n). - Ralf Stephan, Dec 11 2004

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);

Not the same as the period-doubling sequence A096268!

Cf. A073504 and A071992 (curiously A071992 presents the same fractal aspects as Fr(n, m)).

Sequence in context: A129272 A059648 A079261 this_sequence A156729 A049320 A030315

Adjacent sequences: A073056 A073057 A073058 this_sequence A073060 A073061 A073062

nonn

Benoit Cloitre (benoit7848c(AT)orange.fr) and Boris Gourevitch (boris(AT)pi314.net), Aug 16 2002