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

Author

Bradley Klee, Mar 01 2020

Keywords

• nonn

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.

Examples

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

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.

Crossrefs

Cf. A046079, A309507, A000326.

Programs

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]