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A160206 Positive numbers y such that y^2 is of the form x^2+(x+857)^2 with integer x. 3
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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..27.

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.

PROG

(PARI) {forstep(n=-188, 10000000, [3, 1], if(issquare(2*n^2+1714*n+734449, &k), print1(k, ", ")))}

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 A185377

Adjacent sequences:  A160203 A160204 A160205 * A160207 A160208 A160209

KEYWORD

nonn

AUTHOR

Klaus Brockhaus, May 18 2009

STATUS

approved

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Last modified February 20 06:45 EST 2018. Contains 299358 sequences. (Running on oeis4.)