A159811 -id:A159811 - cp's OEIS frontend

A130609 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+223)^2 = y^2.

0, 32, 533, 669, 833, 3672, 4460, 5412, 21945, 26537, 32085, 128444, 155208, 187544, 749165, 905157, 1093625, 4366992, 5276180, 6374652, 25453233, 30752369, 37154733, 148352852, 179238480, 216554192, 864664325, 1044678957, 1262170865, 5039633544, 6088835708

Offset: 1

Mohamed Bouhamida, Jun 17 2007

Also values x of Pythagorean triples (x, x+223, y).
Corresponding values y of solutions (x, y) are in A159809.
For the generic case x^2+(x+p)^2 = y^2 with p = m^2-2 a (prime) number > 7 in A028871, see A118337.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (227+30*sqrt(2))/223 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (105507+65798*sqrt(2))/223^2 for n mod 3 = 0.

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

Cf. A159809, A028871, A118337, A118675, A118676, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159810 (decimal expansion of (227+30*sqrt(2))/223), A159811 (decimal expansion of (105507+65798*sqrt(2))/223^2).

LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,32,533,669,833,3672,4460}, 70]  (* Vladimir Joseph Stephan Orlovsky, Feb 10 2012 *)

{forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+446*n+49729), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+446 for n > 6; a(1)=0, a(2)=32, a(3)=533, a(4)=669, a(5)=833, a(6)=3672.
G.f.: x*(32+501*x+136*x^2-28*x^3-167*x^4-28*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 223*A001652(k) for k >= 0.

Edited and two terms added by Klaus Brockhaus, Apr 30 2009

A159810 Decimal expansion of (227+30*sqrt(2))/223.

Original entry on oeis.org

1, 2, 0, 8, 1, 9, 0, 1, 6, 5, 3, 4, 1, 6, 7, 1, 9, 7, 9, 6, 5, 9, 4, 2, 0, 0, 0, 7, 7, 4, 1, 2, 1, 4, 9, 8, 8, 1, 4, 8, 3, 8, 6, 3, 5, 0, 9, 4, 7, 5, 7, 1, 4, 8, 9, 6, 6, 5, 0, 2, 4, 1, 7, 9, 9, 9, 8, 7, 5, 3, 2, 4, 8, 2, 2, 3, 6, 0, 1, 8, 4, 3, 7, 9, 1, 5, 3, 1, 9, 5, 5, 2, 9, 0, 7, 1, 4, 1, 1, 2, 9, 2, 3, 9, 9

Offset: 1

Klaus Brockhaus, Apr 30 2009

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {1, 2}, b = A130609.

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {0, 2}, b = A159809.

			(227+30*sqrt(2))/223 = 1.20819016534167197965...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A130609, A159809, A002193 (decimal expansion of sqrt(2)), A159811 (decimal expansion of (105507+65798*sqrt(2))/223^2).

(227 +30*Sqrt(2))/223; // G. C. Greubel, May 19 2018

RealDigits[(227+30*Sqrt[2])/223,10,120][[1]] (* Harvey P. Dale, Sep 10 2017 *)

(227 +30*sqrt(2))/223 \\ G. C. Greubel, May 19 2018

Equals (15 +sqrt(2))/(15 -sqrt(2)).

cp's OEIS Frontend

A130609 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+223)^2 = y^2.

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