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A129641
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Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+409)^2 = y^2.
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5
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0, 200, 611, 1227, 2291, 4620, 8180, 14364, 27927, 48671, 84711, 163760, 284664, 494720, 955451, 1660131, 2884427, 5569764, 9676940, 16812660, 32463951, 56402327, 97992351, 189214760, 328737840, 571142264, 1102825427, 1916025531, 3328862051, 6427738620
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OFFSET
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1,2
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COMMENTS
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Also values x of Pythagorean triples (x, x+409, y).
Corresponding values y of solutions (x, y) are in A160577.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (473+168*sqrt(2))/409 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (204819+83570*sqrt(2))/409^2 for n mod 3 = 0.
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LINKS
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FORMULA
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a(n) = 6*a(n-3)-a(n-6)+818 for n > 6; a(1)=0, a(2)=200, a(3)=611, a(4)=1227, a(5)=2291, a(6)=4620.
G.f.: x*(200+411*x+616*x^2-136*x^3-137*x^4-136*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 409*A001652(k) for k >= 0.
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MATHEMATICA
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LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 200, 611, 1227, 2291, 4620, 8180}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)
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PROG
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(PARI) {forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+818*n+167281), print1(n, ", ")))}
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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