A257012 Number of sequences of positive integers with length 5 and alternant equal to n.

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

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: A257008 A338750 A265822 * A325412 A265562 A140312

Adjacent sequences: A257009 A257010 A257011 * A257013 A257014 A257015

Keyword

nonn

Author

Barry R. Smith, Apr 19 2015

Status

approved