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A257012 Number of sequences of positive integers with length 5 and alternant equal to n. 3
0, 0, 1, 2, 3, 5, 5, 10, 8, 11, 11, 19, 15, 19, 17, 27, 17, 36, 17, 43, 27, 29, 31, 54, 30, 41, 45, 63, 29, 57, 33, 75, 49, 59, 47, 96, 39, 79, 57, 84, 61, 81, 49, 97, 81, 85, 47, 150, 64, 105, 75, 101, 69, 123, 77, 141, 81, 103, 71, 189, 75, 119, 121, 137, 82, 143, 85, 183, 101, 129, 93, 211, 89, 129, 131, 187, 116, 201 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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).

LINKS

Table of n, a(n) for n=1..78.

B. R. Smith, Reducing quadratic forms by kneading sequences J. Int. Seq., 17 (2014) 14.11.8.

EXAMPLE

The a(7) = 3 sequences with length 5 and alternant 7 are (1,1,1,3,1), (1,2,1,2,1), and (1,3,1,1,1).

MATHEMATICA

Length5Q[x_, y_] :=

Module[{l = ContinuedFraction[(x[[2]] + 2*x[[1]] + y)/(2*x[[1]])]},

  If[OddQ[Length[l]], Return[Length[l] == 5],

   If[Last[l] == 1, Return[Length[l] - 1 == 5], Return[Length[l] + 1 == 5]]]];

Table[Length[

  Select[Flatten[

    Select[

     Table[{a, k}, {k,

       Select[Range[Ceiling[-Sqrt[n^2 - 4]], Floor[Sqrt[n^2 - 4]]],

        Mod[# - n^2 + 4, 2] == 0 &]}, {a,

       Select[Divisors[(n^2 - 4 - k^2)/4], # > (Sqrt[n^2 - 4] - k)/2 &]}],

     UnsameQ[#, {}] &], 1], Length5Q[#, n] &]], {n, 3, 80}]

CROSSREFS

Cf. A257009, A257010, A257011, A257013, A000012, A000005

Sequence in context: A322770 A257008 A265822 * A325412 A265562 A140312

Adjacent sequences:  A257009 A257010 A257011 * A257013 A257014 A257015

KEYWORD

nonn

AUTHOR

Barry R. Smith, Apr 19 2015

STATUS

approved

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Last modified October 16 21:10 EDT 2019. Contains 328103 sequences. (Running on oeis4.)