cp's OEIS Frontend

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

%I A156650 #15 Mar 17 2023 07:14:52
%S A156650 85,89,91,101,119,145,175,185,221,289,349,371,461,595,769,959,1021,
%T A156650 1241,1649,2005,2135,2665,3451,4469,5579,5941,7225,9605,11681,12439,
%U A156650 15529,20111,26045,32515,34625,42109,55981,68081,72499,90509,117215,151801
%N A156650 Positive numbers y such that y^2 is of the form x^2+(x+119)^2 with integer x.
%C A156650 (-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.
%C A156650 lim_{n -> infinity} a(n)/a(n-9) = 3+2*sqrt(2).
%C A156650 lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))/((9+4*sqrt(2))/7)^2 for n mod 9 = 1.
%C A156650 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}.
%C A156650 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}.
%C A156650 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}.
%C A156650 lim_{n -> infinity} a(n)/a(n-1) = ((9+4*sqrt(2))/7)/((19+6*sqrt(2))/17) for n mod 9 = {5, 6}.
%F A156650 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.
%F A156650 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]
%e A156650 (-51, a(1)) = (-51, 85) is a solution: (-51)^2+(-51+119)^2 = 2601+4624 = 7225 = 85^2.
%e A156650 (A129837(1), a(5)) = (0, 119) is a solution: 0^2+(0+119)^2 = 14161 = 119^2.
%e A156650 (A129837(3), a(7)) = (49, 175) is a solution: 49^2+(49+119)^2 = 2401+28224 = 30625 = 175^2.
%t A156650 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 *)
%o A156650 (PARI) {forstep(n=-52, 120000, [1, 3], if(issquare(n^2+(n+119)^2, &k), print1(k, ",")))}
%Y A156650 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).
%K A156650 nonn
%O A156650 1,1
%A A156650 _Klaus Brockhaus_, Feb 17 2009