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a(n) is the number of sequences (s_1, ..., s_n) of positive integers such that (Product_{k=1..n} [s_k, -1; 1, 0])^2 = [-1, 0; 0, -1].
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%I #21 Apr 13 2022 07:40:27

%S 0,0,2,6,20,75,278

%N a(n) is the number of sequences (s_1, ..., s_n) of positive integers such that (Product_{k=1..n} [s_k, -1; 1, 0])^2 = [-1, 0; 0, -1].

%C Let M(s) denote the matrix

%C [s, -1]

%C [+1, 0]

%C in SL(2,Z). Then we count sequences of positive integers [s_1, ..., s_n] such that (Product_{k=1..n} M(s_k))^2 = -Identity.

%C This is Problem III in the Ovsienko article.

%H Valentin Ovsienko, <a href="https://doi.org/10.1007/s40687-018-0139-z">Partitions of unity in SL(2,Z), negative continued fractions, and dissections of polygons</a>, Res. Math. Sci. 5, 21 (2018); arXiv:<a href="https://arxiv.org/abs/1710.02996">1710.02996</a> [math.CO], 2017-2018.

%Y Cf. A000984 (counts "totally positive" solutions: Sum_k s_k = 3n-3).

%K nonn,more

%O 0,3

%A _Michel Marcus_, Oct 10 2017