In One Sequence Or Another (Non-Fibonacci)

[Marc LeBrun]

 >=Henry Gould
 > I wrote a paper styled "Non-Fibonacci numbers", Fibonacci Quart., 
3(1965), 177-183,
 > about the sequence complementary to 1, 2, 3, 5, 8, 13, 21, 34, . . .

A001690, which references your paper.

 > and showed how a formula can be gotten.
 > ...

Which, alas, is NOT in the OEIS entry.  Could you supply it?


 >=Leroy Quet
 > Do you know what the non-Fibonacci numbers' recursive definition is?

Roughly speaking, a(n+1) = a(n)+1, except when this would produce a 
Fibonacci number, in which case it glitches by +2 instead.

 > Does their direct formula involve powers of (3 + sqrt(5))/2 somehow?
 > (Just a guess. Perhaps the formula does not.)

Well it's basically like n plus the number of Fibonacci numbers less than 
n, to provide the requisite glitches.

Thus you need a formula for the density of Fibonaccis (for complement 
primes we could use the celebrated prime density).

So I think it will involve log(n) or something like that, somehow twiddled, 
roofed and/or summed "in gold".

Alas I don't have time now to take this further; perhaps someone else could 
augment A001690 with a formula?