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Search: id:A065109
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| A065109 |
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Triangle T(n,k) of coefficients relating to Bezier curve continuity. |
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+0
3
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| 1, 2, -1, 4, -4, 1, 8, -12, 6, -1, 16, -32, 24, -8, 1, 32, -80, 80, -40, 10, -1, 64, -192, 240, -160, 60, -12, 1, 128, -448, 672, -560, 280, -84, 14, -1, 256, -1024, 1792, -1792, 1120, -448, 112, -16, 1, 512, -2304, 4608, -5376, 4032, -2016, 672, -144, 18, -1, 1024, -5120, 11520, -15360, 13440
(list; table; graph; listen)
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OFFSET
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0,2
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COMMENT
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sum(binomial(n,i) * (-1)^i * T(i,r), i=0..n) = (-1)^(n-r) * binomial(n,r).
Row sums are 1, antidiagonal sums are the natural numbers. - Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), May 29 2005
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LINKS
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Peter J. Taylor, Conditions for C-a Continuity of Bezier Curves
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FORMULA
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T(n, k) = (-1)^k * 2^(n-k) * binomial(n, k)
For n>0, T(n, k) = 2*T(n-1, k) - T(n-1, k-1). - Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), May 29 2005
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EXAMPLE
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For C-2 continuity between P and Q we require Q_0 = P_n; Q_1 = 2P_n - P_n-1; Q_2 = 4P_n - 4P_n-1 + P_n-2.
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CROSSREFS
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Cf. A038207, A013609. Apart from signs, same as A038207.
Sequence in context: A134397 A134395 A038207 this_sequence A113988 A134308 A117427
Adjacent sequences: A065106 A065107 A065108 this_sequence A065110 A065111 A065112
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KEYWORD
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easy,sign,tabl,nice
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AUTHOR
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Peter J. Taylor, Nov 12 2001
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