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

1,1

Comments

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

The initial values suggest the conjecture that this sequence consists exactly of Catalan(n) 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)

Links

Table of n, a(n) for n=1..49.

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

Example

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.

Prog

(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));

Crossrefs

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

Sequence in context: A206900 A207290 A223066 * A223071 A232431 A267974

Adjacent sequences: A339567 A339568 A339569 * A339571 A339572 A339573

Keyword

nonn,frac

Author

N. J. A. Sloane, Dec 09 2020

Extensions

More terms from Hugo Pfoertner, Dec 09 2020

Status

approved