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%I A365047 #28 Mar 02 2024 13:32:59
%S A365047 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,
%T A365047 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,
%U A365047 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
%N 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.
%C A365047 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.
%H A365047 Scott R. Shannon, <a href="/A365047/b365047.txt">Table of n, a(n) for n = 0..10000</a>
%H A365047 Scott R. Shannon, <a href="/A365047/a365047.png">Image of the first 50000 terms</a>.
%e A365047 a(3) = 1 and a(2) = a(1) = a(0) = 0 form a progression with ratio 1 separated by one term.
%e A365047 a(8) = 1 as a(7) = a(5) = a(3) = 1 for a progression with ratio 1 separated by two terms.
%e A365047 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.
%e A365047 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.
%e A365047 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.
%Y A365047 Cf. A366907 (length>=3), A132345, A365677, A308638, A078651, A051336.
%K A365047 nonn
%O A365047 0,13
%A A365047 _Scott R. Shannon_, Oct 21 2023

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