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A248752
Decimal expansion of limit of the imaginary part of f(1-i,n)/f(1-i,n+1), where f(x,n) is the n-th Fibonacci polynomial.
7
2, 5, 7, 0, 6, 5, 8, 6, 4, 1, 2, 1, 6, 7, 7, 1, 6, 0, 9, 0, 8, 5, 6, 8, 0, 6, 2, 0, 5, 2, 7, 3, 3, 7, 1, 8, 9, 0, 3, 7, 5, 7, 0, 0, 7, 9, 9, 8, 8, 1, 3, 4, 9, 4, 5, 2, 4, 1, 3, 0, 7, 9, 3, 7, 8, 0, 9, 4, 2, 2, 3, 6, 0, 4, 3, 1, 2, 1, 4, 5, 0, 9, 4, 0, 7, 6
OFFSET
0,1
COMMENTS
The analogous limit of f(1,n)/f(1,n+1) is the golden ratio (A001622),
FORMULA
Equals (1-sqrt(sqrt(5)-2))/2. - Vaclav Kotesovec, Oct 19 2014
From Wolfdieter Lang, Mar 02 2018: (Start)
Equals (1 - (2 - phi)*sqrt(phi))/2, with phi = A001622.
Equals (1/10)*y*(1 - (1/50)*y^2) with y = A300070. (End)
EXAMPLE
limit = 0.2570658641216771609085680620527337189037570...
Let q(x,n) = f(x,n)/f(x,n+1) and c = 1-i.
n f(n,x) Re(q(c,n)) Im(q(c,n)))
1 1 1/2 1/2
2 x 3/5 1/5
3 1 + x^2 1/2 1/4
4 2x + x^3 8/15 4/15
5 1 + 3 x^2 + x^4 69/130 33/130
Re(q(11,1+i) = 5021/9490 = 0.5290832...
Im(q(11,1+i) = 4879/18980 = 0.275060...
MAPLE
evalf((1-sqrt(sqrt(5)-2))/2, 120); # Vaclav Kotesovec, Oct 19 2014
MATHEMATICA
z = 300; t = Table[Fibonacci[n, x]/Fibonacci[n + 1, x], {n, 1, z}];
u = t /. x -> 1 - I;
d1 = N[Re[u][[z]], 130]
d2 = N[Im[u][[z]], 130]
r1 = RealDigits[d1] (* A248751 *)
r2 = RealDigits[d2] (* A248752 *)
CROSSREFS
KEYWORD
nonn,easy,cons
AUTHOR
Clark Kimberling, Oct 13 2014
STATUS
approved