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A073059 a(n) = (1/2)*(A073504(n+1) - A073504(n)). 4
0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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).

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384

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)

Index entries for characteristic functions

FORMULA

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.

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

PROG

(PARI) \\ To generate graphs:

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);

(PARI) A073059(n) = if(1==n, 0, if(!(n%2), 0, if(3==(n%4), 1, A073059((n-1)/4)))); \\ Antti Karttunen, Oct 09 2018, after Ralf Stephan's Dec 11 2004 formula

(PARI)

up_to = 10000;

A073504list(up_to) = { my(v=vector(up_to)); for(n=1, up_to, u1=1; u2=n; while((u2!=u1)||((u2%2) == 1), u3=u2; u2=(u2\2)+(u1\2); u1=u3); v[n]=u2); (v); };

v073504 = A073504list(up_to);

A073504(n) = v073504[n];

A073059(n) = (1/2)*(A073504(n+1)-A073504(n)); \\ Antti Karttunen, Nov 27 2018, after code sent by Benoit Cloitre (personal communication), implementing the original definition

CROSSREFS

Not the same as the period-doubling sequence A096268!

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

Cf. also A098725.

Sequence in context: A288707 A079261 A285495 * A288173 A221150 A288997

Adjacent sequences:  A073056 A073057 A073058 * A073060 A073061 A073062

KEYWORD

nonn

AUTHOR

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

EXTENSIONS

Erroneous formula removed by Antti Karttunen, Oct 09 2018

STATUS

approved

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Last modified January 17 16:53 EST 2019. Contains 319235 sequences. (Running on oeis4.)