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

Author

Barry R. Smith, Apr 18 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=5, the a(5)=3 sequences with alternant 5 and length 4 are (1,3,1,1), (1,2,2,1), and (1,1,3,1).
		

Links

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

Crossrefs

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

Programs

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}]