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)).

4, 16, 16, 64, 64, 64, 64, 64, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024

Offset: 1

Author

N. J. A. Sloane, Dec 09 2020

Keywords

• nonn
• frac

Comments

Comment from N. J. A. Sloane, Dec 11 2020: (Start)

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

Hugo Pfoertner tested this conjecture with the PARI program given below.

Here is the output from that program:

[1, 0, 4]

[2, 4, 16]

[4, 16, 64]

[9, 64, 256]

[23, 256, 1024]

[65, 1024, 4096]

[197, 4096, 16384]

[626, 16384, 65536]

[2056, 65536, 262144]

[6918, 262144, 1048576]

[23714, 1048576, 4194304]

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

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

Examples

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

Links

• Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625 [math.CO], 2020-2021. See Section 2.8.
• Raghavendra Tripathi, Proof of conjectured formula

Crossrefs

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

Programs

PARI

\\ Program from Hugo Pfoertner for studying the connection with the Catalan numbers mentioned in the Comments.
a30101(n)=fromdigits(Vecrev(binary(n)), 2);
a62383(n)=1<<(log(2*n+1)\log(2));
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};
A14486=[];for(k=1,5000000,if(is_a14486(k),A14486=concat(A14486,k)));
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));

Extensions

More terms from Hugo Pfoertner, Dec 09 2020