A366907 a(n) is the number of geometric progressions with three or more terms, with rational ratio > 0, formed by the terms a(n-1), a(n-1-k), a(n-1-2*k),...,a(n-1-t*k) where k>=1, t>=2, and n-1-t*k>=0.

0, 0, 0, 1, 0, 1, 0, 2, 0, 4, 1, 0, 1, 0, 0, 2, 0, 3, 0, 3, 0, 6, 0, 7, 0, 9, 0, 13, 0, 12, 0, 15, 0, 21, 0, 20, 0, 22, 0, 30, 0, 30, 0, 31, 0, 38, 0, 39, 0, 43, 0, 47, 0, 46, 0, 53, 0, 61, 0, 57, 0, 59, 0, 69, 0, 72, 0, 72, 0, 78, 0, 79, 0, 84, 0, 91, 0, 90, 0, 96, 0, 103, 0, 98, 0, 105, 0, 116

Offset: 0

Author

Scott R. Shannon, Oct 27 2023

Keywords

• nonn

Comments

The sequence is dominated by the count of progressions consisting of three or more 0's. Very rarely the count of these zero-progressions forms a new progression of its own, which forms a short series of small terms and resets the subsequent count of the zero-progressions to a lower value. In the first 10^5 terms this only happens three times - at a(10) (which is not readily noticeable on the graph of the terms), a(644), and a(61434). See the attached images.

Examples

			a(3) = 1 and a(2) = a(1) = a(0) = 0 form a progression with ratio 1 separated by one term.
a(7) = 2 as a(6) = a(4) = a(2) = 0 form a three-term progression with ratio 1 separated by two terms, while a(6) = a(4) = a(2) = a(0) = 0 form a four-term progression with ratio 1 separated by two terms.
a(10) = 1 as a(9) = 4, a(7) = 2, a(5) = 1 form a three-term progression with ratio 1/2 separated by two terms.
		

Links

• Scott R. Shannon, Table of n, a(n) for n = 0..10000.
• Scott R. Shannon, Image of the first 1000 terms.
• Scott R. Shannon, Image of the first 100000 terms.

Crossrefs

Cf. A365047 (length=3), A132345, A365677, A308638, A078651, A051336.