A319430 First differences of the tribonacci representation numbers (A003726 or A278038).

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 19, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5

Offset: 0

Author

N. J. A. Sloane, Sep 30 2018

Keywords

• nonn

Comments

This sequence appears to consist of runs of 1's of lengths given (essentially) by A275925, separated by single numbers > 1, which define the terms of A319431.

It would be nice to have a recurrence of some kind that produces A319431.

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..50000 (first 9999 terms from N. J. A. Sloane)

Crossrefs

Cf. A000073, A003726, A046630, A089068, A275925, A278038, A319431.

Programs

Mathematica

Differences@ Select[Range[0, 160], SequenceCount[IntegerDigits[#, 2], {1, 1, 1}] == 0 &] (* Michael De Vlieger, Dec 23 2019 *)

Formula

Conjecture: All terms are of the form ceiling(2^k/7) for some k (cf. A046630), and all numbers of the form ceiling(2^k/7) occur.

Conjecture (continued): Furthermore, new values of ceiling(2^k/7) (that is, new records) appear at n = 0, 6,12, 23, 43, 80, 148, 273, ..., which (apart from the start) are the tribonacci numbers minus 1, A000073 - 1, or A089068.

a(n) = ceiling(2^i/7) iff the Tribonacci representation of n+1 ends in i 0's. - Jeffrey Shallit, Oct 02 2018