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A073059 a(n) = (1/2)*(A073504(n+1) - A073504(n)).

%I A073059 #43 May 13 2025 00:52:50
%S A073059 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,
%T A073059 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,
%U A073059 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
%N A073059 a(n) = (1/2)*(A073504(n+1) - A073504(n)).
%C A073059 Let m be any fixed positive integer and let Fr(m,n) := 3*Sum_{k = 1..n} 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(2*p+1) = (5/3)*(4^p - 1) and B(2*p) =(2/3)*(4^p - 1). [Formula for Fr(m,n) corrected by _Petros Hadjicostas_, Oct 21 2019]
%H A073059 Antti Karttunen, <a href="/A073059/b073059.txt">Table of n, a(n) for n = 1..16384</a>
%H A073059 Benoit Cloitre, <a href="/A073059/a073059_4.jpg">Graph of Fr(n,4) for 1 <= n <= B(4)</a>
%H A073059 Benoit Cloitre, <a href="/A073059/a073059_6.jpg">Graph of Fr(n,6) for 1 <= n <= B(6)</a>
%H A073059 Benoit Cloitre, <a href="/A073059/a073059_8.jpg">Graph of Fr(n,8) for 1 <= n <= B(8)</a>
%H A073059 Benoit Cloitre, <a href="/A073059/a073059_5.jpg">Graph of Fr(n,5) for 1 <= n <= B(5)</a>
%H A073059 Benoit Cloitre, <a href="/A073059/a073059_7.jpg">Graph of Fr(n,7) for 1 <= n <= B(7)</a>
%H A073059 Benoit Cloitre, <a href="/A073059/a073059_9.jpg">Graph of Fr(n,9) for 1 <= n <= B(9)</a>
%H A073059 <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>
%F A073059 a(4*k+3) = 1, a(4*k+2) = a(4*k+4) = 0, a(16*k+13) = 1, ...
%F A073059 A073504(n) = Sum_{k = 1..n} a(k) is asymptotic to 2*n/3.
%F A073059 a(2*n) = 0, a(4*n+3) = 1, a(4*n+1) = a(n). - _Ralf Stephan_, Dec 11 2004
%o A073059 (PARI) \\ To generate graphs:
%o A073059 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);
%o A073059 (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
%o A073059 (PARI)
%o A073059 up_to = 10000;
%o A073059 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); };
%o A073059 v073504 = A073504list(up_to);
%o A073059 A073504(n) = v073504[n];
%o A073059 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
%o A073059 (Python)
%o A073059 from functools import lru_cache
%o A073059 @lru_cache(maxsize=None)
%o A073059 def A073059(n): return (1 if n&2 else A073059(n>>2)) if n&1 else 0 # _Chai Wah Wu_, Feb 07 2025
%Y A073059 Not the same as the period-doubling sequence A096268!
%Y A073059 Cf. A073504 and A071992 (curiously A071992 presents the same fractal aspects as Fr(n, m)).
%Y A073059 Cf. A098725.
%K A073059 nonn
%O A073059 1,1
%A A073059 _Benoit Cloitre_ and Boris Gourevitch (boris(AT)pi314.net), Aug 16 2002
%E A073059 Erroneous formula removed by _Antti Karttunen_, Oct 09 2018