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Search: id:A099938
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| A099938 |
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Consider the sequence of circles C0, C1, C2, C3 ..., where C0 is a half-circle of radius 1. C1 is the largest circle that fits into C0 and has radius 1/2. C(n+1) is the largest circle that fits inside C0 but outside C(n), etc. Sequence gives the curvatures (reciprocals of the radii) of the circles. |
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1
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| 2, 4, 18, 100, 578, 3364, 19602, 114244, 665858, 3880900, 22619538, 131836324, 768398402, 4478554084, 26102926098, 152139002500, 886731088898, 5168247530884, 30122754096402, 175568277047524, 1023286908188738
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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The numbers a(2), a(4), a(6) etc. are squares and a(1), a(3), a(5) ... are twice squares. Furthermore, a(1)-2, a(3)-2, a(5)-2 etc. are squares, and a(2)-2, a(4)-2, a(6)-2 etc. are twice square.
C(n) is centered at (x(n), y(n)), where x(n) = sqrt(1-2/a(n)) and y(n) = 1/a(n). - David Wasserman (dwasserm(AT)earthlink.net), Feb 28 2008
C(n) is tangent to C0 because sqrt(x(n)^2+y(n)^2)+y(n) = 1, and C(n) is tangent to C(n+1) because sqrt[(x(n+1)-x(n))^2+(y(n+1)-y(n))^2] = y(n)+y(n+1). - David Wasserman (dwasserm(AT)earthlink.net), Feb 28 2008
a(n+1)/a(n) converges to 3+sqrt(8). - David Wasserman (dwasserm(AT)earthlink.net), Feb 28 2008
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LINKS
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David Wasserman, Illustration of this sequence
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CROSSREFS
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Equals 2 * A055997(n-1).
Sequence in context: A120664 A095816 A020101 this_sequence A135069 A067647 A009667
Adjacent sequences: A099935 A099936 A099937 this_sequence A099939 A099940 A099941
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KEYWORD
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nonn
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AUTHOR
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Hartmut Neubauer (hartmut.f.neubauer(AT)t-online.de), Nov 12 2004
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EXTENSIONS
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More terms from David Wasserman (dwasserm(AT)earthlink.net), Feb 28 2008
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