OFFSET
1,2
COMMENTS
The graph appears to consist of two lines whose slopes are approximately equal to 1.25 and 2.5.
Conjecture from N. J. A. Sloane, Nov 04 2019: (Start)
a(2t) = floor((5t+1)/2) for t >= 1 (essentially A047218),
a(4t+1) = 10t+1(+1 if binary expansion of t ends in odd number of 0's) for t >= 0 (essentially A297469),
a(4t+3) = 10t+7 for t >= 0.
These formulas explain all the known terms.
One could also say that a(4t+1) = 10t+1+A328979(t+1) for t >= 0.
There is a similar conjecture for A328196.
Call the three sets of conjectured terms S0, S1, and S3. The terms in S0 are == 0 or 3 mod 5; those are in S1 are == 1 or 2 mod 10; and those in S3 are == 7 mod 10. So the sets are disjoint, as required by the definition.
This conjecture would imply that the points a(2t) lie on a line of slope 5/4 and the points a(2t+1) on a line of slope 5/2, as conjectured by Peter Kagey. (End)
Comment from N. J. A. Sloane, Nov 06 2019: (Start)
Let us DEFINE a sequence S by the conjectured formulas given here, and a sequence T by the conjectured formulas given in A328196. Then it is not difficult to prove that the first differences of S are given by T, and that the terms of S and T are disjoint.
So S is certainly a candidate for the lexicographically earliest infinite sequence of distinct positive integers such that the sequence and its first differences have no values in common.
Furthermore Peter Kagey's b-files for this sequence and A328196 show that the first 10000 terms of S are indeed the first 10000 terms of the lexicographically earliest such sequence.
But this is not yet a proof that S IS the lexicographically earliest such sequence. (End)
To construct the bisection a(2n-1), start with [4]. Apply the substitution rule 4 -> 46, 5 -> 46, 6 -> 55. Prepend [1, 6] to the resulting list, then take partial sums. - John Keith, Dec 31 2020
LINKS
Peter Kagey, Table of n, a(n) for n = 1..10000
EXAMPLE
a(1) = 1.
a(2) != 1 because a(1) = 1,
a(2) != 2 because then a(2) - a(1) = a(1), so
a(2) = 3.
The first eight terms of this sequence and first seven terms of its first differences are
[1, 3, 7, 5, 11, 8, 17, 10] and
[2, 4, -2, 6, -3, 9, -7] respectively, and these sequences have no common terms.
CROSSREFS
See A328984 and A328985 for simpler sequences which almost have the properties of A329190 and A328196. - N. J. A. Sloane, Nov 07 2019
KEYWORD
nonn
AUTHOR
Peter Kagey, Oct 06 2019
STATUS
approved