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A157297 Positive numbers y such that y^2 is of the form x^2+(x+233)^2 with integer x. 4
185, 233, 317, 793, 1165, 1717, 4573, 6757, 9985, 26645, 39377, 58193, 155297, 229505, 339173, 905137, 1337653, 1976845, 5275525, 7796413, 11521897, 30748013, 45440825, 67154537, 179212553, 264848537, 391405325, 1044527305, 1543650397 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

(-57, a(1)) and (A129625(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+233)^2 = y^2.

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

lim_{n -> infinity} a(n)/a(n-1) = (251+66*sqrt(2))/233 for n mod 3 = {0, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (82611+44030*sqrt(2))/233^2 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)=185, a(2)=233, a(3)=317, a(4)=793, a(5)=1165, a(6)=1717.

G.f.: (1-x)*(185 +418*x +735*x^2 +418*x^3 +185*x^4)/(1-6*x^3+x^6).

a(3*k-1) = 233*A001653(k) for k >= 1.

EXAMPLE

(-57, a(1)) = (-57, 185) is a solution: (-57)^2+(-57+233)^2 = 3249+30976 = 34225 = 185^2.

(A129625(1), a(2)) = (0, 233) is a solution: 0^2+(0+233)^2 = 54289 = 233^2.

(A129625(3), a(4)) = (432, 793) is a solution: 432^2+(432+233)^2 = 186624+442225 = 628849 = 793^2.

MATHEMATICA

LinearRecurrence[{0, 0, 6, 0, 0, -1}, {185, 233, 317, 793, 1165, 1717}, 50] (* G. C. Greubel, Mar 29 2018 *)

PROG

(PARI) {forstep(n=-60, 1100000000, [3, 1], if(issquare(2*n^2+466*n+54289, &k), print1(k, ", ")))};

(MAGMA) I:=[185, 233, 317, 793, 1165, 1717]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, Mar 29 2018

CROSSREFS

Cf. A129625, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A157298 (decimal expansion of (251+66*sqrt(2))/233), A157299 (decimal expansion of (82611+44030*sqrt(2))/233^2).

Sequence in context: A139265 A243628 A262053 * A156059 A244248 A129311

Adjacent sequences:  A157294 A157295 A157296 * A157298 A157299 A157300

KEYWORD

nonn,easy

AUTHOR

Klaus Brockhaus, Apr 11 2009

STATUS

approved

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Last modified April 6 05:38 EDT 2020. Contains 333267 sequences. (Running on oeis4.)