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

%I #22 Jan 27 2020 13:25:39

%S 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,

%T 36,66,0,28,38,40,36,71,12,53,60,57,16,83,14,80,60,32,64,75,50,98,62,

%U 47,16,144,36,100,88,53,52,153,36,94,76,91,98,129,20,92,124,102

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

%C For a guide to related sequences, see A211422.

%C 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

%H Pontus von Brömssen, <a href="/A211510/b211510.txt">Table of n, a(n) for n = 0..1024</a>

%e From _Bernard Schott_, Jan 26 2020: (Start)

%e For n = 4, there are 3 ordered solutions with (1,3,3), (2,3,4) and (2,4,3) so a(4) = 3.

%e For n = 5, there is no solution, hence a(5) = 0.

%e The only solution for n = 6 is (2,4,4) with 2^2 = 4*4 - 2*6, hence a(6) = 1. (End)

%t t[n_] := t[n] = Flatten[Table[w^2 - x*y + 2 n, {w, 1, n}, {x, 1, n}, {y, 1, n}]]

%t c[n_] := Count[t[n], 0]

%t t = Table[c[n], {n, 0, 70}] (* A211510 *)

%o (Python)

%o import sympy

%o 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

%o (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

%Y Cf. A211422.

%K nonn

%O 0,5

%A _Clark Kimberling_, Apr 14 2012

%E Name corrected by _Pontus von Brömssen_, Jan 26 2020