A129857 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+857)^2 = y^2.

0, 235, 1696, 2571, 3796, 12075, 17140, 24255, 72468, 101983, 143448, 424447, 596472, 838147, 2475928, 3478563, 4887148, 14432835, 20276620, 28486455, 84122796, 118182871, 166033296, 490305655, 688822320, 967715035, 2857712848

Offset: 1

Author

Mohamed Bouhamida, Jun 03 2007

Keywords

• nonn
• easy

Comments

Also values x of Pythagorean triples (x, x+857, y).

Corresponding values y of solutions (x, y) are in A160206.

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

lim_{n -> infinity} a(n)/a(n-1) = (907+210*sqrt(2))/857 for n mod 3 = {1, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (1208787+678878*sqrt(2))/857^2 for n mod 3 = 0.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

Crossrefs

Cf. A160206, A001652, A123654, A156035 (decimal expansion of 3+2*sqrt(2)), A160207 (decimal expansion of (907+210*sqrt(2))/857), A160208 (decimal expansion of (1208787+678878*sqrt(2))/857^2).

Programs

Magma

m:=25; R:=PowerSeriesRing(Integers(), m); [0] cat  Coefficients(R!(x*(235+1461*x+875*x^2-185*x^3-487*x^4-185*x^5)/((1-x)*(1-6*x^3+x^6))) );  // G. C. Greubel, May 03 2018

Mathematica

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,235,1696,2571,3796,12075, 17140}, 30] (* or *) CoefficientList[Series[x (235+1461x+875x^2-185x^3- 487x^4- 185x^5)/((1-x)(1-6x^3+x^6)),{x,0,30}],x] (* Harvey P. Dale, Apr 24 2011 *)

PARI

{forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+1714*n+734449), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+1714 for n > 6; a(1)=0, a(2)=235, a(3)=1696, a(4)=2571, a(5)=3796, a(6)=12075.

G.f.: x*(235+1461*x+875*x^2-185*x^3-487*x^4-185*x^5) / ((1-x)*(1-6*x^3+x^6)).

a(3*k+1) = 857*A001652(k) for k >= 0.

Extensions

Edited and two terms added by Klaus Brockhaus, May 18 2009