A161486 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+191)^2 = y^2.

0, 69, 336, 573, 936, 2449, 3820, 5929, 14740, 22729, 35020, 86373, 132936, 204573, 503880, 775269, 1192800, 2937289, 4519060, 6952609, 17120236, 26339473, 40523236, 99784509, 153518160, 236187189, 581587200, 894769869, 1376600280

Offset: 1

Author

Klaus Brockhaus, Jun 13 2009

Keywords

• nonn

Comments

Corresponding values y of solutions (x, y) are in A161487.

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

lim_{n -> infinity} a(n)/a(n-1) = (209+60*sqrt(2))/191 for n mod 3 = {1, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (52323+26522*sqrt(2))/191^2 for n mod 3 = 0.

Links

• Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

Crossrefs

Cf. A161487, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A161488 (decimal expansion of (209+60*sqrt(2))/191), A161489 (decimal expansion of (52323+26522*sqrt(2))/191^2).

Programs

Mathematica

Transpose[NestList[Flatten[{Rest[#],6#[[4]]-First[#]+382}]&,{0,69, 336, 573, 936,2449},40]][[1]]  (* Harvey P. Dale, Apr 01 2011 *)
LinearRecurrence[{1,0,6,-6,0,-1,1},{0,69,336,573,936,2449,3820},40] (* Harvey P. Dale, Mar 29 2016 *)

PARI

{forstep(n=0, 10000000, [1, 3], if(issquare(2*n^2+382*n+36481), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+382 for n > 6; a(1)=0, a(2)=69, a(3)=336, a(4)=573, a(5)=936, a(6)=2449.

G.f.: x*(69+267*x+237*x^2-51*x^3-89*x^4-51*x^5) / ((1-x)*(1-6*x^3+x^6)).

a(3*k+1) = 191*A001652(k) for k >= 0.