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Search: id:A160203
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| A160203 |
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Positive numbers y such that y^2 is of the form x^2+(x+809)^2 with integer x. |
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4
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| 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; listen)
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OFFSET
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1,1
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COMMENT
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(-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.
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FORMULA
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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.
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EXAMPLE
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(-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.
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PROGRAM
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(PARI) {forstep(n=-200, 10000000, [3, 1], if(issquare(2*n^2+1618*n+654481, &k), print1(k, ", ")))}
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CROSSREFS
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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: A043483 A027885 A097105 this_sequence A105130 A117129 A106488
Adjacent sequences: A160200 A160201 A160202 this_sequence A160204 A160205 A160206
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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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