A248752 -id:A248752 - cp's OEIS frontend

A248750 Decimal expansion of limit of the imaginary part of f(1+i,n), where f(x,0) = 1 and f(x,n) = x + 1/f(x,n-1).

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

Offset: 0

Clark Kimberling, Oct 13 2014

See A046854 for a triangle of coefficients of the numerators and denominators of f(x,n). Note that the limit of f(1,n) is the golden ratio.

			0.742934135878322839091431937947266281096242992001186505475869206219...
n   f(x,n)                                 Re(f(1+i,n))  Im(f(1+i,n))
0   1                                      1             0
1   1 + x                                  2             1
2   (1 + x + x^2)/(1 + x)                  7/5           4/5
3   (1 + 2*x + x^2 + x^3)/(1 + x + x^2)    20/13         9/13
Re(f(1+i,10)) = 815/533 = 1.529080...
Im(f(1+i,10)) = 396/533 = 0.742964...

Index entries for algebraic numbers, degree 4.

Cf. A248749, A046854, A248751, A248752, A001622. A300072.

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

$RecursionLimit = Infinity; $MaxExtraPrecision = Infinity;
z = 20; (* For more accuracy, increase z *)
f[x_, n_] := x + 1/f[x, n - 1];
f[x_, 1] = 1; t = Table[Factor[f[x, n]], {n, 1, z}];
u = t /. x -> I + 1; t = Table[Factor[f[x, n]], {n, 1, z}]; u = t /. x -> I + 1;
r1 = N[Re[u][[z]], 130]
r2 = N[Im[u][[z]], 130]
d1 = RealDigits[r1]  (*A248749*)
d2 = RealDigits[r2]  (*A248750*)

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 = -A300072. (End)

A300070 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

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

Offset: 1

Wolfdieter Lang, Mar 02 2018

This number is the second member y of one of the two real triples (x, y, z) which solve the three equations i) x + y + z = 10, ii) x*z = y^2, iii) x^2 + y^2 = z^2. The corresponding numbers are x = 10*A248752 and z = A300071.
The other real solution has x = x2 = 10*A248750, y = y2 = -A300072 and z = z2 = A300073.
The two complex solutions have y3 = 5*(phi + sqrt(phi - 1)*i) with phi = A001622 and i = sqrt(-1), and x3 = y3 - (1/50)*(y3)^3, z3 = 10 - 2*y3  + (1/50)*y3^3.
The polynomial for the solutions Y = y/5 is  P(Y) = Y^4 - 2*Y^3 - 2*Y^2 + 8*Y - 4, or in standard form p(U) = U^4 - (7/2)*U^2 + 5*U - 11/64, with U = Y - 1/2. This factorizes as p(U) = p1(U)*p2(U) with p1(U) = U^2 - (2*phi - 1)*U + 1/4 + phi and p2(U) = U^2 + (2*phi - 1)*U + 5/4 - phi.
This problem appears (see the Havil reference) in Abū Kāmil's Book on Algebra. Havil gives only the positive real solution (x, y, z) on p. 60.
Note that X = x/5, Y = y/5 and Z = z/5 solves i') X + Y + Z = 2, ii) X*Z = Y^2, iii) X^2 + Y^2 = Z^2.

			y = 3.26992830382087058023917813685926686997649431017166693240595879917018...
y/5 = 0.65398566076417411604783562737185337399529886203433338648119175983...

Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 58-60.

I.C. Karpinski, The Algebra of Abu Kamil, Amer. Math. Month. XXI,2 (1914), 37-48.
MacTutor History of Mathematics, Abu Kamil Shuja.
Wikipedia, Abu Kamil.

Cf. A001622, A248750, A248752, A300071, A300072, A300073.

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

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

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

A248751 Decimal expansion of limit of the real part of f(1-i,n)/f(1-i,n+1), where f(x,n) is the n-th Fibonacci polynomial.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Oct 13 2014

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

Differs from A248749 only in the first digit. - R. J. Mathar, Oct 23 2014

			limit = 0.52908551363574612516099052379022521061936504...
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...

Index entries for algebraic numbers, degree 4

Cf. A248750, A248752, A102426, A001622.

evalf((sqrt(2+sqrt(5))-1)/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 *)

polrootsreal(4*x^4+8*x^3+2*x^2-2*x-1)[2] \\ Charles R Greathouse IV, Nov 26 2024

Equals (sqrt(2+sqrt(5))-1)/2. - Vaclav Kotesovec, Oct 19 2014

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.

cp's OEIS Frontend

A248750 Decimal expansion of limit of the imaginary part of f(1+i,n), where f(x,0) = 1 and f(x,n) = x + 1/f(x,n-1).

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A300070 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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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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A248751 Decimal expansion of limit of the real 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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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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