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A257011 Number of sequences of positive integers with length 4 and alternant equal to n. 3
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, 36, 33, 36, 33, 52, 33, 46, 51, 42, 41, 64, 41, 38, 54, 74, 43, 64, 44, 66, 63, 56, 57, 88, 59, 58, 79, 60, 52, 96, 61, 92, 69, 68, 72, 110 (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..60.

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

EXAMPLE

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).

MATHEMATICA

Length4Q[x_, y_] :=

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

  If[EvenQ[Length[l]], Return[Length[l] == 4],

   If[Last[l] == 1, Return[Length[l] - 1 == 4], Return[Length[l] + 1 == 4]]]];

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], Length4Q[#, n] &]], {n, 1, 60}]

CROSSREFS

Cf. A257009, A257010, A257012, A257013, A000012, A000005

Sequence in context: A326136 A276081 A329564 * A144652 A035493 A070038

Adjacent sequences:  A257008 A257009 A257010 * A257012 A257013 A257014

KEYWORD

nonn

AUTHOR

Barry R. Smith, Apr 18 2015

STATUS

approved

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Last modified February 29 09:35 EST 2020. Contains 332355 sequences. (Running on oeis4.)