cp's OEIS Frontend

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.

%I A366907 #10 Oct 28 2023 09:24:43
%S A366907 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,
%T A366907 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,
%U A366907 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
%N 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.
%C A366907 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.
%H A366907 Scott R. Shannon, <a href="/A366907/b366907.txt">Table of n, a(n) for n = 0..10000</a>.
%H A366907 Scott R. Shannon, <a href="/A366907/a366907.png">Image of the first 1000 terms</a>.
%H A366907 Scott R. Shannon, <a href="/A366907/a366907_1.png">Image of the first 100000 terms</a>.
%e A366907 a(3) = 1 and a(2) = a(1) = a(0) = 0 form a progression with ratio 1 separated by one term.
%e A366907 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.
%e A366907 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.
%Y A366907 Cf. A365047 (length=3), A132345, A365677, A308638, A078651, A051336.
%K A366907 nonn
%O A366907 0,8
%A A366907 _Scott R. Shannon_, Oct 27 2023