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

Author

Barry R. Smith, Apr 19 2015

Keywords

• nonn

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).

Examples

			For n=14, the a(14)=4 sequences 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).
		

Links

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

Crossrefs

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

Programs

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}]