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A117691
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Rational numbers in F[n]=(a[m]/b[m])*F[n-1]-F[n-2] that produce harmonic bouncing ball functions from generalized Fibonacci linear recursions.
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0
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4, 3, 3, 2, 8, 5, 5, 3, 12, 7, 7, 4, 16, 9, 9, 5, 20, 11, 11, 6, 24, 13, 13, 7, 28, 15, 15, 8, 32, 17, 17, 9, 36, 19, 19, 10, 40, 21, 21, 11, 44, 23, 23, 12, 48, 25, 25, 13, 52, 27, 27, 14, 56, 29, 29, 15, 60, 31, 31, 16, 64, 33, 33, 17, 68, 35, 35, 18, 72, 37, 37, 19, 76, 39, 39, 20
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| A proper definition is needed.
This method converts a definite sequence of rational numbers into a sequence of Integers. The sequences of the type: f[0] = a0; f[1] = b0; f[n_] := f[n] = (A[m]/B[m])*f[n - 1] - f[n - 2] and M = {{0, 1}, {-1, (A[m]/B[m])}}; v[0] = {a0, bo}; v[n_] := v[n] = M.v[n - 1] are important because they represent an integer based Hilbert space. Because it should be possible to do the equivalent of Fourier expansions in integer recursions using them. Because you can also define orthogonality on integer sequences using them.
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FORMULA
| C[m] = A[m]/B[m], a(n) = {A[m],B[m]}
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MATHEMATICA
| o = Table[Abs[Coefficient[ExpandAll[(x - (a + I*Sqrt[2*a + 1])/(a + 1))*(x - ( a - I*Sqrt[2*a + 1])/(a + 1))], x]], {a, 1, 100}]; rational = Table[{Numerator[o[[n]]], Denominator[o[[n]]]}, {n, 2, 100}]; Flatten[rational]
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CROSSREFS
| Sequence in context: A120927 A117323 A016502 * A171627 A143487 A031350
Adjacent sequences: A117688 A117689 A117690 * A117692 A117693 A117694
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KEYWORD
| nonn,uned,obsc,less
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 12 2006
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