cp's OEIS Frontend

A339570 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)).

%I A339570 #42 Mar 09 2025 17:03:29
%S A339570 4,16,16,64,64,64,64,64,256,256,256,256,256,256,256,256,256,256,256,
%T A339570 256,256,256,1024,1024,1024,1024,1024,1024,1024,1024,1024,1024,1024,
%U A339570 1024,1024,1024,1024,1024,1024,1024,1024,1024,1024,1024,1024,1024,1024,1024,1024
%N A339570 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 A339570 Comment from _N. J. A. Sloane_, Dec 11 2020: (Start)
%C A339570 The initial values suggest the conjecture that this sequence consists exactly of Catalan(k) copies of 4^k for k >= 1.
%C A339570 _Hugo Pfoertner_ tested this conjecture with the PARI program given below.
%C A339570 Here is the output from that program:
%C A339570   [1, 0, 4]
%C A339570   [2, 4, 16]
%C A339570   [4, 16, 64]
%C A339570   [9, 64, 256]
%C A339570   [23, 256, 1024]
%C A339570   [65, 1024, 4096]
%C A339570   [197, 4096, 16384]
%C A339570   [626, 16384, 65536]
%C A339570   [2056, 65536, 262144]
%C A339570   [6918, 262144, 1048576]
%C A339570   [23714, 1048576, 4194304]
%C A339570 The first column is A014137, the partial sums of the Catalan numbers, which is strong support for the conjecture.
%C A339570 The conjecture has now been proved by _Raghavendra Tripathi_ - see link. (End)
%H A339570 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 A339570 Raghavendra Tripathi, <a href="/A339570/a339570.pdf">Proof of conjectured formula</a>
%e A339570 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 A339570 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 A339570 a(n) is the denominator of b(n), and A072800(n) is the numerator.
%o A339570 (PARI) \\ Program from _Hugo Pfoertner_ for studying the connection with the Catalan numbers mentioned in the Comments.
%o A339570 a30101(n)=fromdigits(Vecrev(binary(n)), 2);
%o A339570 a62383(n)=1<<(log(2*n+1)\log(2));
%o A339570 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 A339570 A14486=[];for(k=1,5000000,if(is_a14486(k),A14486=concat(A14486,k)));
%o A339570 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 A339570 Cf. A000108, A014137, A030101, A062383, A072800.
%K A339570 nonn,frac
%O A339570 1,1
%A A339570 _N. J. A. Sloane_, Dec 09 2020
%E A339570 More terms from _Hugo Pfoertner_, Dec 09 2020