A211510 Number of ordered triples (w,x,y) with all terms in {1,...,n} and w^2 = x*y - 2n.

0, 0, 0, 0, 3, 0, 1, 6, 5, 4, 13, 0, 16, 12, 7, 8, 22, 10, 27, 20, 20, 8, 41, 14, 27, 32, 21, 36, 66, 0, 28, 38, 40, 36, 71, 12, 53, 60, 57, 16, 83, 14, 80, 60, 32, 64, 75, 50, 98, 62, 47, 16, 144, 36, 100, 88, 53, 52, 153, 36, 94, 76, 91, 98, 129, 20, 92, 124, 102

Offset: 0

Author

Clark Kimberling, Apr 14 2012

Keywords

• nonn

Comments

For a guide to related sequences, see A211422.

The original name was "... and w^2 = x*y + 2n", but this would yield 2 instead of 0 for a(3), as observed by Pontus von Brömssen. The corresponding sequence seems not to be in the OEIS yet. - M. F. Hasler, Jan 26 2020

Examples

			From _Bernard Schott_, Jan 26 2020: (Start)
For n = 4, there are 3 ordered solutions with (1,3,3), (2,3,4) and (2,4,3) so a(4) = 3.
For n = 5, there is no solution, hence a(5) = 0.
The only solution for n = 6 is (2,4,4) with 2^2 = 4*4 - 2*6, hence a(6) = 1. (End)
		

Links

• Pontus von Brömssen, Table of n, a(n) for n = 0..1024

Crossrefs

Cf. A211422.

Programs

Mathematica

t[n_] := t[n] = Flatten[Table[w^2 - x*y + 2 n, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* A211510 *)

PARI

apply( {A211510(n)=sum(w=1,n-2,my(w2n=(w^2-1)\n+2,s); fordiv(w^2+2*n,x, x>w2n||next; x>n&&break; s++);s)}, [1..100]) \\ M. F. Hasler, Jan 26 2020

Python

import sympy
def A211510(n): return sum(x<=n and x*n>=w**2+2*n for w in range(1,n+1) for x in sympy.divisors(w**2+2*n)) # Pontus von Brömssen, Jan 26 2020

Extensions

Name corrected by Pontus von Brömssen, Jan 26 2020