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A160203 Positive numbers y such that y^2 is of the form x^2+(x+809)^2 with integer x. 3
641, 809, 1105, 2741, 4045, 5989, 15805, 23461, 34829, 92089, 136721, 202985, 536729, 796865, 1183081, 3128285, 4644469, 6895501, 18232981, 27069949, 40189925, 106269601, 157775225, 234244049, 619384625, 919581401, 1365274369 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

(-200, a(1)) and (A123654(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+809)^2 = y^2.

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

lim_{n -> infinity} a(n)/a(n-1) = (873+232*sqrt(2))/809 for n mod 3 = {0, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (989043+524338*sqrt(2))/809^2 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)=641, a(2)=809, a(3)=1105, a(4)=2741, a(5)=4045, a(6)=5989.

G.f.: (1-x)*(641+1450*x+2555*x^2+1450*x^3+641*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 809*A001653(k) for k >= 1.

EXAMPLE

(-200, a(1)) = (-200, 641) is a solution: (-200)^2+(-200+809)^2 = 40000+370881 = 410881 = 641^2.

(A123654(1), a(2)) = (0, 809) is a solution: 0^2+(0+809)^2 = 654481 = 809^2.

(A123654(3), a(4)) = (1491, 2741) is a solution: 1491^2+(1491+809)^2 = 2223081+5290000 = 7513081 = 2741^2.

PROG

(PARI) {forstep(n=-200, 10000000, [3, 1], if(issquare(2*n^2+1618*n+654481, &k), print1(k, ", ")))}

CROSSREFS

Cf. A123654, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160204 (decimal expansion of (873+232*sqrt(2))/809), A160205 (decimal expansion of (989043+524338*sqrt(2))/809^2).

Sequence in context: A252426 A256777 A252425 * A251840 A252433 A252434

Adjacent sequences:  A160200 A160201 A160202 * A160204 A160205 A160206

KEYWORD

nonn

AUTHOR

Klaus Brockhaus, May 18 2009

STATUS

approved

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Last modified January 20 17:22 EST 2018. Contains 297960 sequences.