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A259055
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a(n) = 9*n^2 + 18*n + 7.
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1
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7, 34, 79, 142, 223, 322, 439, 574, 727, 898, 1087, 1294, 1519, 1762, 2023, 2302, 2599, 2914, 3247, 3598, 3967, 4354, 4759, 5182, 5623, 6082, 6559, 7054, 7567, 8098, 8647, 9214, 9799, 10402, 11023, 11662, 12319, 12994, 13687, 14398, 15127
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OFFSET
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0,1
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COMMENTS
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a(n) gives twice the curvature of the n-th circle touching the two semicircles of the (2/3,1/3) arbelos and the (n-1)-th circle, with input circle of twice the curvature a(0) = A114949(1) = 7 (referring to the second circle of the counterclockwise Pappus chain).
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LINKS
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FORMULA
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a(n) = 9*(n+1)^2 - 2, n >= 0.
O.g.f.: (-2*x^2+13*x +7)/(1-x)^3.
Recurrence: a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), n >= 3, with a(0) = 7, a(1)= 34 and a(2) = 79.
Descartes' three (actually five) circle theorem (see links) leads to a nonlinear recurrence for twice the curvatures: a(n) = 2*(3 + 3/2) + a(n-1) + 4*sqrt((3 + 3/2)*a(n-1)/2 + 9/2) = 9 + a(n-1) + 6*sqrt(a(n-1) + 2), with input a(0) = 7 = 2*A114949(1). This leads to a quadratic equation with the relevant solution a(n) = 9*n^2 + 18*n + 7.
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MAPLE
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MATHEMATICA
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LinearRecurrence[{3, -3, 1}, {7, 34, 79}, 50] (* Harvey P. Dale, Sep 05 2018 *)
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PROG
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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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STATUS
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approved
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