A257010 Number of sequences of positive integers with length 3 and alternant equal to n.

0, 2, 2, 4, 3, 6, 2, 9, 4, 6, 5, 11, 4, 9, 6, 10, 5, 14, 2, 16, 7, 6, 9, 16, 6, 11, 8, 17, 5, 14, 4, 20, 10, 8, 9, 22, 2, 17, 10, 16, 11, 14, 6, 18, 13, 12, 5, 28, 6, 19, 9, 15, 13, 16, 8, 24, 6, 12, 11, 32, 6, 15, 16, 16, 9, 19, 8, 30, 8, 14, 9, 30, 8, 15, 12, 21, 16, 22

Offset

3,2

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).

There are infinitely many sequences of length 3 and alternant equal to 2. It is for this reason that the offset is 3.

Links

Andrew Howroyd, Table of n, a(n) for n = 3..1000

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

Formula

a(n) = Sum_{b=1..n-1} (Dbm (b,b^2+nb+1)-2), where Dbm(b,m) is the number of positive divisors of m that are congruent to 1 modulo b. - Barry R. Smith, Jan 24 2016

Example

For n=6, the a(6) = 4 sequences with alternant 6 are (1,1,3), (1,3,2), (2,3,1), (3,1,1)

Maple

Dbm:= proc(b, m) nops(select(t -> (t-1) mod b = 0, numtheory:-divisors(m))) end proc:
seq(add(Dbm(b, b^2+n*b+1)-2, b=1..n-1), n=3..100); # Robert Israel, Jan 24 2016

Mathematica

Length3Q[x_, y_] :=
Module[{l = ContinuedFraction[(x[[2]] + 2*x[[1]] + y)/(2*x[[1]])]},
  If[OddQ[Length[l]], Return[Length[l] == 3],
   If[Last[l] == 1, Return[Length[l] - 1 == 3], Return[Length[l] + 1 == 3]]]];
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], Length3Q[#, n] &]], {n, 3, 60}]

Prog

(PARI) a(n)={sum(b=1, n-1, sumdiv(b^2+n*b+1, d, (d-1)%b==0) - 2)} \\ Andrew Howroyd, May 01 2020

Crossrefs

Cf. A257009, A257011, A257012, A257013, A000012, A000005.

Sequence in context: A275735 A328835 A076435 * A156864 A059975 A087656

Adjacent sequences: A257007 A257008 A257009 * A257011 A257012 A257013

Keyword

nonn

Author

Barry R. Smith, Apr 18 2015

Extensions

Terms a(61) and beyond from Andrew Howroyd, May 01 2020

Status

approved