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COMMENTS
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For all such b(i) sequences there is a value of i = Nmax, such that for i<=Nmax the sequence produces integers and for i > Nmax the sequence produces rational non-integers. Thus the "length" of the integer portion of b(i) is Nmax + 1 (when including b(0)).
Nmax is a function of the integer k > 0, written as Nmax(k).
Let B(i,k) be the array of all b(i) sequences, where k also serves as column index.
By definition k=1 is record-breaking. Nmax(1) = 3.
B(i,1) has four integer elements: 1,1,3,6.
Nmax(2k) = 1 for all k. Only b(0) and b(1) are integers when k is even.
A record-breaking b(i) has k = r such that Nmax(r) > Nmax(k) for all k < r.
Nmax(k) varies in an irregular pattern, tied to powers of 2 as explained below. This is also true of b(i) sequence families with other nonzero values of b(0).
(This contrasts with the family where b(0) = 0. In that case the record-breaking values k are simply the powers of 2, since each record-breaking Nmax is +1 greater in length than the prior such Nmax.)
Here, the values of Nmax for record-breaking b(i) lengths starting at k = 1 are 3, 6, 7, 8, 9, 12, 13, 15, 17, 18, 19, 20, 22, 23, 25, 28, 31, 32, 33, 39, 41, 43, 44, 47, 50, 54, 56, 59, 60, 62, 63, 64, 65, 69, 70, 71, 72, 73, 74, 76, 80, 83, 87, 89, 90, ...
B(i,k) follows these rules, regardless of irregularity in Nmax:
Rule 1: If B(i,k) is an integer then B(i, k + 2^(i-1)) is an integer.
Rule 2: If k = r1 and k = r2 yield two consecutive record-breaking b(i) sequences, then r2 = r1 + 2^(Nmax(r1) - 1).
Thus the index of the next record-breaking sequence can be found without searching.
Rule 3: B(i,k + 2^(i-1)) - B(i,k) = A083858(i) for all k, regardless of whether the B(i,k) sequence elements selected are integers.
Conjecture: Rules 1 & 2 apply for all such b(i) sequences of the form: b(i) = m*(b(i-1) + b(i-2))/2, b(0) = j, b(1) = k, for j >= 0 and odd integers m > 2.
Rule 3, for any odd m, generalizes as: B(i,k + 2^(i-1)) - B(i,k), for all k, equals sequences of the form f(i) = m*f(i-1) + 2m*f(i-2), f(0) = 0, f(1) = 1.
It appears that a simple rule for finding the next record-breaking sequence may not exist for b(i) sequences of the form: b(i) = 2*(b(i-1) + b(i-2))/3, b(0) = 1, b(1) = 1. A factor 2 must be "occasionally" applied to the predicted power of 3.
Also note, (a(n)-1)/4 is an integer. The first seven nonzero values of this reduced form are the same as A113828, but otherwise no relationship.
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