A257013 Number of sequences of positive integers with length 6 and alternant equal to n.

0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 5, 0, 4, 4, 9, 0, 12, 1, 13, 10, 8, 4, 33, 4, 14, 12, 21, 4, 44, 2, 33, 22, 24, 12, 62, 8, 16, 29, 63, 2, 64, 4, 57, 52, 26, 10, 111, 21, 40, 48, 45, 8, 106, 26, 94, 40, 46, 18, 164, 21, 40, 61, 97, 40, 118, 12, 87, 65, 104, 14, 221, 14, 52, 116, 88, 30, 146, 21, 157

Offset

1,10

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..80.

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

Example

For n=14, the a(14)=4 sequences of with alternant 14 and length 6 are (1,1,1,1,4,1), (1,2,1,1,3,1), (1,3,1,1,2,1), and (1,4,1,1,1,1).

Mathematica

Length6Q[x_, y_] :=
 Module[{l = ContinuedFraction[(x[[2]] + 2*x[[1]] + y)/(2*x[[1]])]},
  If[EvenQ[Length[l]], Return[Length[l] == 6],
   If[Last[l] == 1, Return[Length[l] - 1 == 6], Return[Length[l] + 1 == 6]]]]
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], Length6Q[#, n] &]], {n, 1, 80}]

Crossrefs

Cf. A257009, A257010, A257011, A257012, A000012, A000005

Sequence in context: A369743 A374941 A369907 * A104755 A242690 A054013

Adjacent sequences: A257010 A257011 A257012 * A257014 A257015 A257016

Keyword

nonn

Author

Barry R. Smith, Apr 19 2015

Status

approved