A330657 Number of ways the n-th pentagonal number A000326(n) can be written as the difference of two positive pentagonal numbers.

0, 0, 0, 1, 1, 0, 2, 0, 0, 1, 0, 1, 1, 1, 0, 1, 2, 0, 2, 1, 1, 3, 1, 0, 2, 3, 0, 3, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 2, 1, 1, 1, 1, 3, 2, 2, 2, 1, 4, 1, 0, 2, 1, 2, 1, 1, 1, 3, 2, 2, 1, 3, 1, 3, 2, 1, 6, 1, 1, 1, 3, 2

Offset

1,7

Comments

Equivalently, a(n) is the number of triples [n,k,m] with k>0 satisfying the Diophantine equation n*(3*n-1) + k*(3*k-1) - m*(3*m-1) = 0. Any such triple satisfies a triangle inequality, n+k > m. The n for which there is a triple [n,n,m] are listed in A137694. Solutions of the form [n,m-1,m] appear only when n=3*z+1, z > 0. The n for which a(n)=0 are listed in A135768.

References

N. J. A. Sloane et al., "sum of 2 triangular numbers is a triangular number", math-fun mailing list, Feb. 19-29, 2020.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000

B. Klee, Pentathagorean triples: two easy proofs, seqfan mailing list, Mar. 20, 2020.

M. A. Nyblom, On the representation of the integers as a difference of nonconsecutive triangular numbers, Fibonacci Quarterly 39:3 (2001), pp. 256-263.

Example

Isosceles case, n=5: 2*5*(3*5-1) - 7*(3*7-1) = 0.

Mathematica

PentaTriples[PNn_] := Sort[Select[{PNn,
      (-PNn + 3 PNn^2 + # - 3 #^2)/(6 #),
      (-PNn + 3 PNn^2 + # + 3 #^2)/(6 #)
      } & /@ Divisors[PNn*(3*PNn - 1)],
   And[And @@ (IntegerQ /@ #), And @@ (# > 0 & /@ #)] &]]
Length[PentaTriples[#]] & /@ Range[100]
a[n_] := Length@FindInstance[n > 0 && y > 0 && z > 0 &&
     n (3 n - 1) + y (3 y - 1) == z (3 z - 1), {y, z}, Integers, 10^9];
a /@ Range[100]

Crossrefs

Cf. A046079, A309507, A000326.

Sequence in context: A234578 A069859 A179317 * A307598 A324967 A320332

Adjacent sequences: A330654 A330655 A330656 * A330658 A330659 A330660

Keyword

nonn

Author

Bradley Klee, Mar 01 2020

Status

approved