A365047 a(n) is the number of three-term geometric progressions, with rational ratio > 0, formed by the terms a(n-1), a(n-1-k) and a(n-1-2*k), where k >= 1 and n - 1 - 2*k >= 0.

0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 2, 0, 0, 2, 0, 3, 0, 4, 2, 0, 0, 4, 1, 0, 1, 0, 2, 1, 0, 3, 0, 5, 0, 4, 1, 0, 2, 0, 2, 0, 5, 0, 4, 1, 3, 0, 4, 1, 1, 1, 2, 1, 4, 2, 0, 4, 1, 0, 3, 0, 3, 0, 2, 2, 1, 4, 0, 5, 0, 3, 0, 6, 0, 3, 1, 3, 0, 5, 0, 6, 0, 5, 0, 6, 0, 6, 0, 8, 0, 8, 0, 9, 1, 2, 1, 1, 2

Offset

0,13

Comments

The sequence is dominated by the count of three-term progressions consisting of three 0's. The 0 terms alternate between long runs on the even and odd n values, so the larger nonzero terms alternate between counting the progressions on these two subsequences, leading to two interrupted lines on the graph of the terms, along with the much lower counts of other three-term subsequences. See the attached image.

Links

Scott R. Shannon, Table of n, a(n) for n = 0..10000

Scott R. Shannon, Image of the first 50000 terms.

Example

a(3) = 1 and a(2) = a(1) = a(0) = 0 form a progression with ratio 1 separated by one term.
a(8) = 1 as a(7) = a(5) = a(3) = 1 for a progression with ratio 1 separated by two terms.
a(12) = 2 as a(11) = a(8) = a(5) = 1 form a progression with ratio 1 separated by three terms, while a(11) = a(7) = a(3) = 1 form a progression with ratio 1 separated by four terms.
a(20) = 2 as a(19) = 4, a(15) = 2, a(11) = 1 form a progression with ratio 1/2 separated by four terms, while a(19) = 4, a(12) = 2, a(5) = 1  form a progression with ratio 1/2 separated by seven terms.
a(170) = 1 as a(169) = 16, a(131) = 12, a(93) = 9 form a progression with ratio 3/4 separated by thirty-eight terms. This is the first series with a ratio that is not an integer or an integer reciprocal.

Crossrefs

Cf. A366907 (length>=3), A132345, A365677, A308638, A078651, A051336.

Sequence in context: A334596 A291900 A263146 * A361015 A028959 A317642

Adjacent sequences: A365044 A365045 A365046 * A365048 A365049 A365050

Keyword

nonn

Author

Scott R. Shannon, Oct 21 2023

Status

approved