A300070 -id:A300070 - cp's OEIS frontend

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.

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

Clark Kimberling, Oct 13 2014

The analogous limit of f(1,n)/f(1,n+1) is the golden ratio (A001622).

			limit = 0.2570658641216771609085680620527337189037570...
Let q(x,n) = f(x,n)/f(x,n+1) and c = 1-i.
n   f(x,n)            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   2*x + x^3         8/15         4/15
5   1 + 3*x^2 + x^4   69/130       33/130
Re(q(1-i,11)) = 5021/9490 = 0.5290832...
Im(q(1-i,11)) = 4879/18980 = 0.257060...

Cf. A248750, A248751, A102426, A001622, A300070.

evalf((1-sqrt(sqrt(5)-2))/2, 120); # Vaclav Kotesovec, Oct 19 2014

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 *)

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)

A300071 Decimal expansion of the member z of a triple (x, y, z) solving a certain historical system of three equations with positive y.

Original entry on oeis.org

4, 1, 5, 9, 4, 1, 3, 0, 5, 4, 9, 6, 2, 3, 5, 7, 8, 1, 0, 6, 7, 5, 1, 4, 1, 2, 4, 2, 6, 1, 3, 3, 9, 5, 9, 4, 0, 9, 8, 5, 9, 3, 5, 6, 0, 9, 8, 4, 0, 1, 9, 8, 1, 2, 2, 3, 5, 2, 7, 3, 3, 2, 6, 3, 0, 2, 0, 3, 9, 2, 0, 7, 5, 9, 8, 2, 9, 5, 7, 6, 7, 9, 9, 7, 6, 4, 7, 1, 5, 7, 5, 1, 5, 4, 4, 1, 5, 0, 0, 9

Offset: 1

Wolfdieter Lang, Mar 02 2018

See A300071 for the system of equations and references and links.
The current triple is (x = 10*A248752, A300070, z = this entry).
The other real solution is (x = x2 = 10*A248750, y = y2 = A300072, z = z2 = A300073).

			z = 4.15941305496235781067514124261339594098593560984019812235273326302039...
z/5 = 0.831882610992471562135028248522679188197187121968039624470546652604...

Cf. A001622, A248750, A248752, A300070, A300072, A300073.

RealDigits[5 (GoldenRatio - (GoldenRatio - 1) Sqrt[GoldenRatio]), 10, 100][[1]] (* Bruno Berselli, Mar 02 2018 *)

z = 5*(phi - (phi-1)*sqrt(phi)), with the golden section phi = (1 + sqrt(5))/2 = A001622.

z = 10 - 2*y + (1/50)*y^3, with y = A300070.

A300072 Decimal expansion of the positive member -y of a triple (x, y, z) solving a certain historical system of three equations.

Original entry on oeis.org

9, 4, 5, 0, 2, 6, 8, 1, 9, 1, 3, 1, 9, 8, 1, 9, 0, 6, 2, 2, 8, 5, 0, 4, 6, 4, 8, 0, 5, 1, 5, 6, 4, 8, 0, 4, 7, 1, 7, 9, 5, 8, 6, 1, 0, 8, 2, 2, 9, 2, 9, 5, 5, 5, 3, 7, 6, 0, 4, 4, 5, 0, 2, 6, 2, 2, 2, 7, 9, 0, 1, 9, 1, 7, 7, 4, 8, 5, 2, 3, 0, 7, 6, 8, 7, 9, 5, 7, 0, 9, 5, 8, 8, 9, 2, 5, 6, 9, 8

Offset: 1

Wolfdieter Lang, Mar 02 2018

The system of three equations is
  x + y + z = 10,
  x*z = y^2,
  x^2 + y^2 = z^2.
See A300070 for the Havil reference and links to Abū Kāmil, who considered this system. This real solution was not given in Havil's book.
This solution is x = x2 := 10*A248750, -y = -y2 = present entry, z = z2 = A300073.
The other real solution with positive y is x = 10*A248752, y = A300070, z = A300071.
Note that X2 = x2/5, -Y2 = -y2/5 and Z2 = z2/5 solve the system of equations (i) X2 + Y2 + Z2 = 2, (ii) X2*Z2 = (Y2)^2 and (iii) (X2)^2 + (Y2)^2 = (Z2)^2.

			-y2 = 9.450268191319819062285046480515648047179586108229295553760445026222...
-y2/5 = 1.8900536382639638124570092961031296094359172216458591107520890052...

Cf. A001622, A248750, A248752, A300070, A300071, A300073.

RealDigits[5 (1 - GoldenRatio - Sqrt[GoldenRatio]), 10, 100][[1]] (* Bruno Berselli, Mar 02 2018 *)

-y2 = 5*(1 - phi - sqrt(phi)), with the golden section phi = (1 + sqrt(5))/2 = A001622.

The minimal polynomial is x^4 + 10*x^3 - 50*x^2 - 1000*x - 2500. - Joerg Arndt, Jul 21 2025

A300073 Decimal expansion of the member z of a triple (x, y, z) satisfying a certain historical system of three equations with negative y.

Original entry on oeis.org

1, 2, 0, 2, 0, 9, 2, 6, 8, 3, 2, 5, 3, 6, 5, 9, 0, 6, 7, 1, 3, 7, 0, 7, 2, 7, 1, 0, 1, 0, 4, 2, 9, 8, 5, 2, 3, 6, 2, 1, 7, 1, 5, 6, 1, 8, 8, 2, 1, 7, 4, 3, 0, 4, 9, 9, 0, 0, 1, 7, 5, 2, 9, 6, 4, 0, 3, 2, 2, 1, 2, 5, 5, 2, 2, 0, 6, 0, 6, 6, 8, 1, 7, 0, 9, 5, 6, 0, 0, 4, 6, 6, 7, 3, 9, 4, 9, 6, 3, 6

Offset: 2

Wolfdieter Lang, Mar 02 2018

See A300070 and A300072 for the system of equations, the Havil reference and links to Abū Kāmil.

The present solution is x = x2 = 10*A248750, -y = -y2 = A300072, z = z2 = present entry.

			z2 = 12.02092683253659067137072710104298523621715618821743049900175296403...
z2/5 = 2.4041853665073181342741454202085970472434312376434860998003505928...

Cf. A001622, A248750, A248752, A300070, A300071, A300072.

RealDigits[5 (GoldenRatio + (GoldenRatio - 1) Sqrt[GoldenRatio]), 10, 100][[1]] (* Bruno Berselli, Mar 02 2018 *)

z2 = 5*(phi + (phi - 1)*sqrt(phi)), with the golden section phi = (1 + sqrt(5))/2 = A001622.

z2 = 10 - 2*y2 + (1/50)*y2^3, with y2 = -A300072.

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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.

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A300071 Decimal expansion of the member z of a triple (x, y, z) solving a certain historical system of three equations with positive y.

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A300072 Decimal expansion of the positive member -y of a triple (x, y, z) solving a certain historical system of three equations.

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A300073 Decimal expansion of the member z of a triple (x, y, z) satisfying a certain historical system of three equations with negative y.

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