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A160090 Positive numbers y such that y^2 is of the form x^2 + (x + 569)^2 with integer x. 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 23 16:46 EDT 2019. Contains 328373 sequences. (Running on oeis4.)