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A222465
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a(n) = 4*n^2 + 3.
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4
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3, 7, 19, 39, 67, 103, 147, 199, 259, 327, 403, 487, 579, 679, 787, 903, 1027, 1159, 1299, 1447, 1603, 1767, 1939, 2119, 2307, 2503, 2707, 2919, 3139, 3367, 3603, 3847, 4099, 4359, 4627, 4903, 5187, 5479, 5779, 6087, 6403, 6727, 7059, 7399
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OFFSET
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0,1
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COMMENTS
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2/a(n) = R(n)/r, n >= 0, with R(n) the n-th radius of the clockwise Pappus chain of the arbelos with semicircle radii r, r1 = 2r/3, r2 = r/3. See the MathWorld link for Pappus chain (there only the counterclockwise chain is shown). The counterclockwise chain companion has circle radii R(n)/r = 2/A114949(n), n>=0.
Binomial transform of (3, 4, 8, 0, 0, 0, 0, 0, 0, 0, ...). - Philippe Deléham, Mar 07 2013
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LINKS
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FORMULA
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a(n) = 4*n^2 + 3, n >= 0.
O.g.f.: (3 - 2*x + 7*x^2)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2, a(0) = 3, a(1) = 7, a(2) = 19. - Philippe Deléham, Mar 05 2013
Sum_{n>=0} 1/a(n) = 1/6 + sqrt(3)*Pi*coth(sqrt(3)*Pi/2)/12.
Sum_{n>=0} (-1)^n/a(n) = 1/6 + sqrt(3)*Pi*cosech(sqrt(3)*Pi/2)/12. (End)
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EXAMPLE
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The dimensionless radii R(n)/r of the clockwise Pappus chain for the arbelos (r,r1,r2=r-r1) = r*(1,2/3,1/3) are [2/3, 2/7, 2/19, 2/39, 2/67, 2/103, 2/147, 2/199 ,...], for n >= 0. The circle for n=0 has radius r1=2/3 and center (2/3,0) with the origin at the left tip of the arbelos. The n=1 circle coincides with the one of the counterclockwise companion chain.
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MAPLE
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MATHEMATICA
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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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