%I A117691
%S A117691 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,
%T A117691 17,17,9,36,19,19,10,40,21,21,11,44,23,23,12,48,25,25,13,52,27,27,14,56,
%U A117691 29,29,15,60,31,31,16,64,33,33,17,68,35,35,18,72,37,37,19,76,39,39,20
%N 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.
%C A117691 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.
%C A117691 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]
%F A117691 C[m]=A[m]/B[m] a(n) = {A[m],B[m]}
%t A117691 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]
%Y A117691 Sequence in context: A120927 A117323 A016502 this_sequence A143487 A031350 
               A031353
%Y A117691 Adjacent sequences: A117688 A117689 A117690 this_sequence A117692 A117693 
               A117694
%K A117691 nonn,uned,probation
%O A117691 0,1
%A A117691 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 12 2006