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Denote the van der Corput sequence of fractions 1/2, 1/4, 3/4, 1/8, 5/8, 3/8, 7/8, 1/16, ... (A030101/A062383) by v(n), n >= 1. Then a(n) = denominator of v(A014486(n)).
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%I #38 Nov 24 2023 17:17:35

%S 4,16,16,64,64,64,64,64,256,256,256,256,256,256,256,256,256,256,256,

%T 256,256,256,1024,1024,1024,1024,1024,1024,1024,1024,1024,1024,1024,

%U 1024,1024,1024,1024,1024,1024,1024,1024,1024,1024,1024,1024,1024,1024,1024,1024

%N Denote the van der Corput sequence of fractions 1/2, 1/4, 3/4, 1/8, 5/8, 3/8, 7/8, 1/16, ... (A030101/A062383) by v(n), n >= 1. Then a(n) = denominator of v(A014486(n)).

%C Comment from _N. J. A. Sloane_, Dec 11 2020: (Start)

%C The initial values suggest the conjecture that this sequence consists exactly of Catalan(n) copies of 4^k for k >= 1.

%C _Hugo Pfoertner_ tested this conjecture with the PARI program given below.

%C Here is the output from that program:

%C [1, 0, 4]

%C [2, 4, 16]

%C [4, 16, 64]

%C [9, 64, 256]

%C [23, 256, 1024]

%C [65, 1024, 4096]

%C [197, 4096, 16384]

%C [626, 16384, 65536]

%C [2056, 65536, 262144]

%C [6918, 262144, 1048576]

%C [23714, 1048576, 4194304]

%C The first column is A014137, the partial sums of the Catalan numbers, which is strong support for the conjecture.

%C The conjecture has now been proved by Raghavendra Tripathi - see link. (End)

%H Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, <a href="https://arxiv.org/abs/2012.04625">Finding structure in sequences of real numbers via graph theory: a problem list</a>, arXiv:2012.04625 [math.CO], 2020-2021. See Section 2.8.

%H Raghavendra Tripathi, <a href="/A339570/a339570.pdf">Proof of conjectured formula</a>

%e The van der Corput sequence v(n), n >= 1, is 1/2, 1/4, 3/4, 1/8, 5/8, 3/8, 7/8, 1/16, 9/16, 5/16, 13/16, 3/16, 11/16, ... = A030101/A062383.

%e Then we construct the sequence b(n) = v(A014486(n)), n >= 1, which is 1/4, 5/16, 3/16, 21/64, 13/64, 19/64, 11/64, 7/64, ...

%e a(n) is the denominator of b(n), and A072800(n) is the numerator.

%o (PARI) \\ Program from _Hugo Pfoertner_ for studying the connection with the Catalan numbers mentioned in the Comments.

%o a30101(n)=fromdigits(Vecrev(binary(n)), 2);

%o a62383(n)=1<<(log(2*n+1)\log(2));

%o is_a14486(n)={my(v=binary(n), t=0); for(i=1, #v, t+=if(v[i], 1, -1); if(t<0, return(0))); t==0};

%o A14486=[];for(k=1,5000000,if(is_a14486(k),A14486=concat(A14486,k)));

%o aprev=0;for(k=1,#A14486,my(j=A14486[k],a=denominator(a30101(j)/a62383(j)));if(a!=aprev,print([k,aprev,a]);aprev=a));

%Y Cf. A000108, A014137, A030101, A062383, A072800.

%K nonn,frac

%O 1,1

%A _N. J. A. Sloane_, Dec 09 2020

%E More terms from _Hugo Pfoertner_, Dec 09 2020