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%I #6 Apr 19 2015 01:00:55
%S 0,0,1,2,3,6,5,8,11,10,10,20,11,16,21,24,15,26,23,28,31,22,24,49,27,
%T 36,33,36,33,52,33,46,51,42,41,64,41,38,54,74,43,64,44,66,63,56,57,88,
%U 59,58,79,60,52,96,61,92,69,68,72,110
%N Number of sequences of positive integers with length 4 and alternant equal to n.
%C See A257009 for the definition of the alternant of a sequence. The number of sequences of length 1 with given alternant value n is 1, while the number of sequences of length 2 with given alternant value n is d(n), the number of divisors of n (see A000005).
%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.
%e For n=5, the a(5)=3 sequences of with alternant 5 and length 4 are (1,3,1,1),(1,2,2,1), and (1,1,3,1).
%t Length4Q[x_, y_] :=
%t Module[{l = ContinuedFraction[(x[[2]] + 2*x[[1]] + y)/(2*x[[1]])]},
%t If[EvenQ[Length[l]], Return[Length[l] == 4],
%t If[Last[l] == 1, Return[Length[l] - 1 == 4], Return[Length[l] + 1 == 4]]]];
%t Table[Length[
%t Select[Flatten[
%t Select[
%t Table[{a, k}, {k,
%t Select[Range[Ceiling[-Sqrt[n^2 + 4]], Floor[Sqrt[n^2 + 4]]],
%t Mod[# - n^2 - 4, 2] == 0 &]}, {a,
%t Select[Divisors[(n^2 + 4 - k^2)/4], # > (Sqrt[n^2 + 4] - k)/2 &]}],
%t UnsameQ[#, {}] &], 1], Length4Q[#, n] &]], {n, 1, 60}]
%Y Cf. A257009, A257010, A257012, A257013, A000012, A000005
%K nonn
%O 1,4
%A _Barry R. Smith_, Apr 18 2015
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