A160090 Positive numbers y such that y^2 is of the form x^2 + (x + 569)^2 with integer x.

485, 569, 689, 2221, 2845, 3649, 12841, 16501, 21205, 74825, 96161, 123581, 436109, 560465, 720281, 2541829, 3266629, 4198105, 14814865, 19039309, 24468349, 86347361, 110969225, 142611989, 503269301, 646776041, 831203585, 2933268445

Offset

1,1

Comments

(-93, a(1)) and (A101152(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+569)^2 = y^2.

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

Lim_{n -> infinity} a(n)/a(n-1) = (587+102*sqrt(2))/569 for n mod 3 = {0, 2}.

Lim_{n -> infinity} a(n)/a(n-1) = (617139+371510*sqrt(2))/569^2 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)=485, a(2)=569, a(3)=689, a(4)=2221, a(5)=2845, a(6)=3649.

G.f.: (1-x)*(485 +1054*x +1743*x^2 +1054*x^3 +485*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 569*A001653(k) for k >= 1.

Example

(-93, a(1)) = (-93, 485) is a solution: (-93)^2+(-93+569)^2 = 8649+226576 = 235225 = 485^2.
(A101152(1), a(2)) = (0, 569) is a solution: 0^2+(0+569)^2 = 323761= 569^2.
(A101152(3), a(4)) = (1260, 2221) is a solution: 1260^2+(1260+569)^2 = 1587600+3345241 = 4932841 = 2221^2.

Mathematica

LinearRecurrence[{0, 0, 6, 0, 0, -1}, {485, 569, 689, 2221, 2845, 3649}, 50] (* G. C. Greubel, Apr 21 2018 *)

Prog

(PARI) {forstep(n=-96, 10000000, [3, 1], if(issquare(2*n^2+1138*n+323761, &k), print1(k, ", ")))}
(PARI) x='x+O('x^30); Vec((1-x)*(485 +1054*x +1743*x^2 +1054*x^3 +485*x^4)/(1-6*x^3+x^6)) \\ G. C. Greubel, Apr 21 2018
(Magma) I:=[485, 569, 689, 2221, 2845, 3649]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, Apr 21 2018

Crossrefs

Cf. A101152, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160091 (decimal expansion of (587+102*sqrt(2))/569), A160092 (decimal expansion of (617139+371510*sqrt(2))/569^2).

Sequence in context: A013771 A114774 A013903 * A158326 A031722 A156774

Adjacent sequences: A160087 A160088 A160089 * A160091 A160092 A160093

Keyword

nonn

Author

Klaus Brockhaus, May 04 2009

Status

approved