A156650 Positive numbers y such that y^2 is of the form x^2+(x+119)^2 with integer x.

85, 89, 91, 101, 119, 145, 175, 185, 221, 289, 349, 371, 461, 595, 769, 959, 1021, 1241, 1649, 2005, 2135, 2665, 3451, 4469, 5579, 5941, 7225, 9605, 11681, 12439, 15529, 20111, 26045, 32515, 34625, 42109, 55981, 68081, 72499, 90509, 117215, 151801

Offset: 1

Author

Klaus Brockhaus, Feb 17 2009

Keywords

• nonn

Comments

(-51, a(1)), (-39, a(2)), (-35, a(3)), (-20, a(4)) and (A129837(n), a(n+4)) are solutions (x, y) to the Diophantine equation x^2+(x+119)^2 = y^2.

lim_{n -> infinity} a(n)/a(n-9) = 3+2*sqrt(2).

lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))/((9+4*sqrt(2))/7)^2 for n mod 9 = 1.

lim_{n -> infinity} a(n)/a(n-1) = ((9+4*sqrt(2))/7)^2*((19+6*sqrt(2))/17)/(3+2*sqrt(2)) for n mod 9 = {0, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))/(((9+4*sqrt(2))/7)*((19+6*sqrt(2))/17)^2) for n mod 9 = {3, 8}.

lim_{n -> infinity} a(n)/a(n-1) = ((19+6*sqrt(2))/17)^2/((9+4*sqrt(2))/7) for n mod 9 = {4, 7}.

lim_{n -> infinity} a(n)/a(n-1) = ((9+4*sqrt(2))/7)/((19+6*sqrt(2))/17) for n mod 9 = {5, 6}.

Examples

			(-51, a(1)) = (-51, 85) is a solution: (-51)^2+(-51+119)^2 = 2601+4624 = 7225 = 85^2.
(A129837(1), a(5)) = (0, 119) is a solution: 0^2+(0+119)^2 = 14161 = 119^2.
(A129837(3), a(7)) = (49, 175) is a solution: 49^2+(49+119)^2 = 2401+28224 = 30625 = 175^2.
		

Crossrefs

Cf. A129837, A156035 (decimal expansion of 3+2*sqrt(2)), A156649 (decimal expansion of (9+4*sqrt(2))/7), A156163 (decimal expansion of (19+6*sqrt(2))/17).

Programs

Mathematica

upto=200000; With[{max=Ceiling[(Sqrt[2*upto^2]-119)/2]},Union[ Sqrt[#]&/@ Select[Table[x^2+(x+119)^2,{x,-250,max}],IntegerQ[Sqrt[#]]&]]](* Harvey P. Dale, Aug 11 2011 *)

PARI

{forstep(n=-52, 120000, [1, 3], if(issquare(n^2+(n+119)^2, &k), print1(k, ",")))}

Formula

a(n) = 6*a(n-9)-a(n-18) for n > 18; a(1)=85, a(2)=89, a(3)=91, a(4)=101, a(5)=119, a(6)=145, a(7)=175, a(8)=185, a(9)=221, a(10)=289, a(11)=349, a(12)=371, a(13)=461, a(14)=595, a(15)=769, a(16)=959, a(17)=1021, a(18)=1241.

G.f.: x * (1-x) * (85 +174*x +265*x^2 +366*x^3 +485*x^4 +630*x^5 +805*x^6 +990*x^7 +1211*x^8 +990*x^9 +805*x^10 +630*x^11 +485*x^12 +366*x^13 +265*x^14 +174*x^15 +85*x^16) / (1 -6*x^9 +x^18). [adapted to the offset by Bruno Berselli, Apr 01 2011]