A160206 Positive numbers y such that y^2 is of the form x^2+(x+857)^2 with integer x.

697, 857, 1117, 3065, 4285, 6005, 17693, 24853, 34913, 103093, 144833, 203473, 600865, 844145, 1185925, 3502097, 4920037, 6912077, 20411717, 28676077, 40286537, 118968205, 167136425, 234807145, 693397513, 974142473, 1368556333

Offset

1,1

Comments

(-185, a(1)) and (A129857(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+857)^2 = y^2.

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 = {0, 2}.

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

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

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

Formula

a(n) = 6*a(n-3) -a(n-6) for n > 6; a(1)=697, a(2)=857, a(3)=1117, a(4)=3065, a(5)=4285, a(6)=6005.

G.f.: (1-x)*(697+1554*x+2671*x^2+1554*x^3+697*x^4)/(1-6*x^3+x^6).

a(3*k-1) = 857*A001653(k) for k >= 1.

Example

(-185, a(1)) = (-185, 697) is a solution: (-185)^2+(-185+857)^2 = 34225+451584 = 485809 = 697^2.
(A129857(1), a(2)) = (0, 857) is a solution: 0^2+(0+857)^2 = 734449 = 857^2.
(A129857(3), a(4)) = (1696, 3065) is a solution: 1696^2+(1696+857)^2 = 2876416+6517809 = 9394225 = 3065^2.

Mathematica

LinearRecurrence[{0, 0, 6, 0, 0, -1}, {697, 857, 1117, 3065, 4285, 6005}, 50] (* G. C. Greubel, May 14 2018 *)

Prog

(PARI) {forstep(n=-188, 10000000, [3, 1], if(issquare(2*n^2 +1714*n +734449, &k), print1(k, ", ")))}
(PARI) x='x+O('x^30); Vec((1-x)*(697+1554*x+2671*x^2+1554*x^3 +697*x^4 )/(1-6*x^3+x^6)) \\ G. C. Greubel, May 14 2018
(Magma) I:=[697, 857, 1117, 3065, 4285, 6005]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..40]]; // G. C. Greubel, May 14 2018

Crossrefs

Cf. A129857, A001653, 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).

Sequence in context: A282549 A069330 A218156 * A111105 A137559 A306097

Adjacent sequences: A160203 A160204 A160205 * A160207 A160208 A160209

Keyword

nonn

Author

Klaus Brockhaus, May 18 2009

Status

approved