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%I #12 Apr 19 2015 01:00:37
%S 4,8,9,17,14,25,22,36,25,49,31,55,49,69,41,83,52,100,66,100,66,126,84,
%T 132,88,125,95,198,82,159,119,190,125,211,125,194,135,275,128,250,152,
%U 232,191,238,174,348,150,330,223,279,158,356,220,374,217,360,196,438
%N Number of finite sequences of positive integers with alternant equal to n.
%C The alternant of a sequence of positive integers (c_1, ..., c_r) with r>=3 is the positive integer [c_1, ..., c_r] - [c_2, ..., c_{r-1}], in which an expression in brackets denotes the numerator of the simplified rational number with continued fraction expansion having the sequence of quotients in brackets. The alternant of (c_1) is c_1 and the alternant of (c_1, c_2) is c_1*c_2. There are finitely many sequences with given alternant >= 3. (There are infinitely many sequences with alternant 2 -- (2), (1,2), (2,1), and all sequences of the form (1,p,1). It is for this reason that the offset is 3.)
%C The number of Zagier-reduced binary quadratic forms with discriminant equal to n^2-4 or n^2+4
%C The number of pairs of integers (h,k) with |k| < sqrt(D), k^2 congruent to D (mod 4), h > (sqrt(D) - k)/2, h exactly dividing (D-k^2)/4, where D=n^2+4 or n^2-4.
%C a(n) = A257007(n) + A257008(n)
%D D. B. Zagier, Zetafunktionen und quadratische Korper, Springer, 1981.
%H B. R. Smith, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Smith/smith5.html">Reducing quadratic forms by kneading sequences</a> J. Int. Seq., 17 (2014) 14.11.8.
%F a(n) equals the number of pairs (h,k) with |k| < sqrt(D), k^2 congruent to D (mod 4), h > (sqrt(D) - k)/2, h exactly dividing (D-k^2)/4, where D = n^2-4 or n^2+4.
%Y Cf. A257003, A257007, A257008
%K nonn
%O 3,1
%A _Barry R. Smith_, Apr 16 2015
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