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Search: id:A160209
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| A160209 |
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Positive numbers y such that y^2 is of the form x^2+(x+937)^2 with integer x. |
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4
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| 673, 937, 1685, 2353, 4685, 9437, 13445, 27173, 54937, 78317, 158353, 320185, 456457, 922945, 1866173, 2660425, 5379317, 10876853, 15506093, 31352957, 63394945, 90376133, 182738425, 369492817, 526750705, 1065077593, 2153561957
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OFFSET
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1,1
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COMMENT
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(-385, a(1)) and (A129974(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+937)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (1179+506*sqrt(2))/937 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (933747+224782*sqrt(2))/937^2 for n mod 3 = 1.
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FORMULA
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a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=673, a(2)=937, a(3)=1685, a(4)=2353, a(5)=4685, a(6)=9437.
G.f.: (1-x)*(673+1610*x+3295*x^2+1610*x^3+673*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 937*A001653(k) for k >= 1.
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EXAMPLE
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(-385, a(1)) = (-385, 673) is a solution: (-385)^2+(-385+937)^2 = 148225+304704 = 452929 = 673^2.
(A129974(1), a(2)) = (0, 937) is a solution: 0^2+(0+937)^2 = 877969 = 937^2.
(A129974(3), a(4)) = (1128, 2353) is a solution: 1128^2+(1128+937)^2 = 1272384+4264225 = 5536609 = 2353^2.
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PROGRAM
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(PARI) {forstep(n=-388, 10000000, [3, 1], if(issquare(2*n^2+1874*n+877969, &k), print1(k, ", ")))}
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CROSSREFS
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Cf. A129974, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160210 (decimal expansion of (1179+506*sqrt(2))/937), A160211 (decimal expansion of (933747+224782*sqrt(2))/937^2).
Sequence in context: A057797 A057802 A047728 this_sequence A158392 A124942 A158393
Adjacent sequences: A160206 A160207 A160208 this_sequence A160210 A160211 A160212
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KEYWORD
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nonn
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AUTHOR
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Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), May 18 2009
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