|
Search: id:A117691
|
|
|
| A117691 |
|
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. |
|
+0
1
|
|
| 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; listen)
|
|
|
OFFSET
|
0,1
|
|
|
COMMENT
|
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. Becuse 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.
a(n)= mix A022998(n+2) , A026741(n+3) . A026741 has many links to Bernoulli numbers (A027641/A027642 or A164555/A027642) and Rydberg-Ritz spectra of hydrogen,in particular Balmer A061037. A022998 is linked to Balmer A061038;see A145979. A026741(n-1)+A026741(n+1)=A022998. A022998(n+1)/A026741(n+1)=period 2:repeat 1,4=A010685 . [From Paul Curtz (bpcrtz(AT)free.fr), Sep 19 2009]
|
|
FORMULA
|
C[m]=A[m]/B[m] a(n) = {A[m],B[m]}
|
|
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]
|
|
CROSSREFS
|
Sequence in context: A120927 A117323 A016502 this_sequence A143487 A031350 A031353
Adjacent sequences: A117688 A117689 A117690 this_sequence A117692 A117693 A117694
|
|
KEYWORD
|
nonn,uned,probation
|
|
AUTHOR
|
Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 12 2006
|
|
|
Search completed in 0.002 seconds
|
