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A130013
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Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+647)^2 = y^2.
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5
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0, 37, 1768, 1941, 2128, 11937, 12940, 14025, 71148, 76993, 83316, 416245, 450312, 487165, 2427616, 2626173, 2840968, 14150745, 15308020, 16559937, 82478148, 89223241, 96519948, 480719437, 520032720, 562561045, 2801839768, 3030974373
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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+647, y).
Corresponding values y of solutions (x, y) are in A159641.
For the generic case x^2+(x+p)^2 = y^2 with p = 2*m^2-1 a (prime) number in A066436 see A118673 or A129836.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (649+36*sqrt(2))/647 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (1084467+707402*sqrt(2))/647^2 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)+1294 for n > 6; a(1)=0, a(2)=37, a(3)=1768, a(4)=1941, a(5)=2128, a(6)=11937.
G.f.: x*(37+1731*x+173*x^2-35*x^3-577*x^4-35*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 647*A001652(k) for k >= 0.
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PROG
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(PARI) {forstep(n=0, 10000000, [1, 3], if(issquare(2*n^2+1294*n+418609), 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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