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Search: id:A160041
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| A160041 |
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Positive numbers y such that y^2 is of the form x^2+(x+73)^2 with integer x. |
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4
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| 53, 73, 125, 193, 365, 697, 1105, 2117, 4057, 6437, 12337, 23645, 37517, 71905, 137813, 218665, 419093, 803233, 1274473, 2442653, 4681585, 7428173, 14236825, 27286277, 43294565, 82978297, 159036077, 252339217, 483632957, 926930185
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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(-28, a(1)) and (A129289(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+73)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (89+36*sqrt(2))/73 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (5907+1802*sqrt(2))/73^2 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)=53, a(2)=73, a(3)=125, a(4)=193, a(5)=365, a(6)=697.
G.f.: (1-x)*(53+126*x+251*x^2+126*x^3+53*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 73*A001653(k) for k >= 1.
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EXAMPLE
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(-28, a(1)) = (-28, 53) is a solution: (-28)^2+(-28+73)^2 = 784+2025 = 2809 = 53^2.
(A129289(1), a(2)) = (0, 73) is a solution: 0^2+(0+73)^2 = 5329 = 73^2.
(A129289(3), a(4)) = (95, 193) is a solution: 95^2+(95+73)^2 = 9025+28224 = 37249 = 193^2.
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PROGRAM
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(PARI) {forstep(n=-28, 10000000, [3, 1], if(issquare(2*n^2+146*n+5329, &k), print1(k, ", ")))}
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CROSSREFS
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Cf. A129289, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160042 (decimal expansion of (89+36*sqrt(2))/73), A160043 (decimal expansion of (5907+1802*sqrt(2))/73^2).
Sequence in context: A136073 A133187 A057667 this_sequence A107309 A039389 A043212
Adjacent sequences: A160038 A160039 A160040 this_sequence A160042 A160043 A160044
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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 04 2009
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