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A159893
Positive numbers y such that y^2 is of the form x^2+(x+727)^2 with integer x.
4
677, 727, 785, 3277, 3635, 4033, 18985, 21083, 23413, 110633, 122863, 136445, 644813, 716095, 795257, 3758245, 4173707, 4635097, 21904657, 24326147, 27015325, 127669697, 141783175, 157456853, 744113525, 826372903, 917725793
OFFSET
1,1
COMMENTS
(-52, a(1)) and (A130646(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+727)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (731+54*sqrt(2))/727 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (1304787+843542*sqrt(2))/727^2 for n mod 3 = 1.
For the generic case x^2+(x+p)^2=y^2 with p= m^2 -2 a prime number in A028871, m>=5, the x values are given by the sequence defined by: a(n)= 6*a(n-3) -a(n-6) +2*p with a(1)= 0, a(2)= 2*m +2, a(3)= 3*m^2 -10*m +8, a(4)= 3*p, a(5)= 3*m^2 +10*m +8, a(6)= 20*m^2 -58*m +42. Y values are given by the sequence defined by: b(n)= 6*b(n-3) -b(n-6) with b(1)= p, b(2)= m^2 +2*m +2, b(3)= 5*m^2 -14*m +10, b(4)= 5*p, b(5)= 5*m^2 +14*m +10, b(6)= 29*m^2 -82*m +58. - Mohamed Bouhamida, Sep 09 2009
FORMULA
a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=677, a(2)=727, a(3)=785, a(4)=3277, a(5)=3635, a(6)=4033.
G.f.: (1-x)*(677+1404*x+2189*x^2+1404*x^3+677*x^4)/(1-6*x^3+x^6).
a(3*k-1) = 727*A001653(k) for k >= 1.
EXAMPLE
(-52, a(1)) = (-52, 677) is a solution: (-52)^2+(-52+727)^2 = 2704+455625 = 458329 = 677^2.
(A130646(1), a(2)) = (0, 727) is a solution: 0^2+(0+727)^2 = 528529 = 727^2.
(A130646(3), a(4)) = (1925, 3277) is a solution: 1925^2+(1925+727)^2 = 3705625+7033104 = 10738729 = 3277^2.
MATHEMATICA
LinearRecurrence[{0, 0, 6, 0, 0, -1}, {677, 727, 785, 3277, 3635, 4033}, 50] (* G. C. Greubel, May 17 2018 *)
PROG
(PARI) {forstep(n=-52, 10000000, [1, 3], if(issquare(2*n^2+1454*n+528529, &k), print1(k, ", ")))}
(Magma) I:=[677, 727, 785, 3277, 3635, 4033]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, May 17 2018
CROSSREFS
Cf. A130646, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159894 (decimal expansion of (731+54*sqrt(2))/727), A159895 (decimal expansion of (1304787+843542*sqrt(2))/727^2).
Sequence in context: A014759 A058462 A058450 * A142755 A158386 A031730
KEYWORD
nonn,easy
AUTHOR
Klaus Brockhaus, Apr 30 2009
STATUS
approved