A257011 Number of sequences of positive integers with length 4 and alternant equal to n.

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

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 A363277

Adjacent sequences: A257008 A257009 A257010 * A257012 A257013 A257014

Keyword

nonn

Author

Barry R. Smith, Apr 18 2015

Status

approved