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

1,1

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}.

Links

Table of n, a(n) for n=1..42.

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]

Example

(-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.

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 *)

Prog

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

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).

Sequence in context: A003906 A215432 A020312 * A044978 A308516 A095593

Adjacent sequences: A156647 A156648 A156649 * A156651 A156652 A156653

Keyword

nonn

Author

Klaus Brockhaus, Feb 17 2009

Status

approved