A002390 -id:A002390 - cp's OEIS frontend

A139341 Decimal expansion of e^((1+sqrt(5))/2).

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

Offset: 1

Mohammad K. Azarian, Apr 14 2008

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 13 2019

			5.04316564336002865131188218928542471032359017541384...

Index entries for transcendental numbers.

Cf. A001622, A094214, A104457, A098317, A002390, A139339, A139340, A139342.

phi := (1+sqrt(5))/2 ; evalf(exp(phi)) ; # R. J. Mathar, Oct 16 2015

RealDigits[Exp[GoldenRatio], 10, 100][[1]] (* Amiram Eldar, Feb 08 2022 *)

exp((sqrt(5)+1)/2) \\ Charles R Greathouse IV, May 13 2019

From Amiram Eldar, Feb 08 2022: (Start)
Equals exp(A001622).
Equals 1/A139342. (End)

A153386 Decimal expansion of Sum_{n>=1} 1/Fibonacci(2*n).

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Dec 25 2008

			1.535370508836252985029852896651599006367...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.14.1, p. 358.

Richard André-Jeannin, Problem B-745, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 31, No. 3 (1993), p. 277; Fun with Unit Fractions, Solution to Problem B-745 by Paul S. Bruckman, ibid., Vol. 33, No. 1 (1995), p. 86.
A. F. Horadam, Elliptic Functions and Lambert Series in the Summation of Reciprocals in Certain Recurrence-Generated Sequences, The Fibonacci Quarterly, Vol. 26, No. 2 (1988), pp. 98-114.

Cf. A000045, A001906 (Fibonacci(2*n)), A065563.

Cf. A000796, A001113, A002163, A002390, A016628, A079586, A153387, A256178, A265288, A360928.

rd[k_] := rd[k] = RealDigits[ N[ Sum[ 1/Fibonacci[2*n], {n, 1, 2^k}], 105]][[1]]; rd[k = 4]; While[ rd[k] != rd[k - 1], k++]; rd[k] (* Jean-François Alcover, Oct 29 2012 *)
RealDigits[Sqrt[5] * (Log[5] + 2*QPolyGamma[0, 1, 1/GoldenRatio^4] - 4*QPolyGamma[0, 1, 1/GoldenRatio^2]) / (8*ArcCsch[2]), 10, 105][[1]] (* Vaclav Kotesovec, Feb 26 2023 *)

sumpos(n=1, 1/fibonacci(2*n)) \\ Michel Marcus, Sep 04 2021

Equals sqrt(5) * (L((3-sqrt(5))/2) - L((7-3*sqrt(5))/2)), where L(x) = Sum_{k>=1} x^k/(1-x^k) (Horadam, 1988, equation (4.6)). - Amiram Eldar, Oct 04 2020
From Gleb Koloskov, Sep 04 2021: (Start)
Equals 1/2 + (sqrt(5)/log(phi))*(log(5)/8 + 3*Integral_{x=0..infinity} sin(x)/((4*sin(x)^2+5)*(exp(Pi*x/log(phi))-1)) dx), where phi = (1+sqrt(5))/2 = A001622.
Equals 1/2 + (A002163/A002390)*(A016628/8 + 3*Integral_{x=0..infinity} sin(x)/((4*sin(x)^2+5)*(A001113^(A000796*x/A002390)-1)) dx). (End)
Equals 1 + Sum_{n>=1} 1/A065563(2*n-1) (André-Jeannin, 1993). - Amiram Eldar, Jan 15 2022
From Peter Bala, Aug 17 2022: (Start)
Equals 5/3 - 3*Sum_{n >= 1} 1/(F(2*n)*F(2*n+2)*F(2*n+4)), where F(n) = Fibonacci(n).
Conjecture: Equals 151/96 - 6*Sum_{n >= 1} 1/(F(2*n)*F(2*n+4)*F(2*n+6)). (End)
Equals A360928 * sqrt(5). - Kevin Ryde, Feb 27 2023

A263401 Expansion of Product_{k>=1} (1 + x^k - x^(2*k)).

Original entry on oeis.org

1, 1, 0, 2, 0, 1, 3, 1, 1, 2, 6, 1, 4, 2, 5, 10, 5, 4, 9, 7, 8, 21, 9, 13, 13, 19, 13, 27, 32, 23, 29, 33, 27, 45, 37, 45, 79, 49, 57, 68, 82, 67, 101, 83, 109, 155, 124, 113, 174, 148, 171, 196, 215, 198, 262, 310, 269, 330, 314, 342, 414, 430, 393, 536, 493

Offset: 0

Vaclav Kotesovec, Jan 03 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A000726, A002390, A055922, A100847, A109389, A162891, A266686.

nmax = 80; CoefficientList[Series[Product[1+x^k-x^(2*k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; p = ConstantArray[0, nmax + 1]; p[[1]] = 1; p[[2]] = 1; p[[3]] = -1; Do[Do[p[[j+1]] = p[[j+1]] + p[[j - k + 1]] - If[j < 2*k, 0, p[[j - 2*k + 1]]], {j, nmax, k, -1}];, {k, 2, nmax}]; p (* Vaclav Kotesovec, May 10 2018 *)

a(n) ~ sqrt(log(phi)) * phi^sqrt(8*n) / (2^(3/4)*sqrt(Pi)*n^(3/4)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jan 03 2016

A139340 Decimal expansion of the cube root of the golden ratio. That is, the decimal expansion of ((1+sqrt(5))/2)^(1/3).

Original entry on oeis.org

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

Offset: 1

Mohammad K. Azarian, Apr 14 2008

Larger of the real roots of x^6 - x^3 - 1. - Charles R Greathouse IV, Apr 14 2014

			1.1739849967053285...

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Index entries for algebraic numbers, degree 6

Cf. A001622, A094214, A104457, A098317, A002390, A139339.

phi := (sqrt(5)+1)/2 ; evalf(root[3](phi)) ; # R. J. Mathar, Oct 16 2015

RealDigits[N[GoldenRatio^(1/3),200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 06 2012 *)

polrootsreal(x^6-x^3-1)[2] \\ Charles R Greathouse IV, Apr 14 2014

A290565 Decimal expansion of sum of reciprocal golden rectangle numbers.

Original entry on oeis.org

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

Offset: 1

Bobby Jacobs and Robert G. Wilson v, Aug 06 2017

The constant k in A277266 such that A277266(n) ~ k*n.

			1/(1*1) + 1/(1*2) + 1/(2*3) + 1/(3*5) + ... = 1 + 1/2 + 1/6 + 1/15 + ... = 1.77387758328513234380...

Cf. A000045, A000796, A001622, A001654, A002163, A002390, A079586, A094214, A265290, A277266.

RealDigits[ Sum[1/(Fibonacci[k]*Fibonacci[k + 1]), {k, 265}], 10, 111][[1]]

suminf(n=1, 1/(fibonacci(n)*fibonacci(n+1))) \\ Michel Marcus, Feb 19 2019

Equals Sum_{n>=1} 1/(Fibonacci(n)*Fibonacci(n+1)).
Equals lim_{n->infinity} A277266(n)/n.
Equals 2 * (Sum_{k>=1} 1/(phi^k * F(k))) - 1/phi = 2 * A265290 - A094214, where phi is the golden ratio (A001622) and F(k) is the k-th Fibonacci number (A000045). - Amiram Eldar, Oct 05 2020
Equals 3/2 + 10*c*Integral_{x=0..infinity} f(x) dx, where c = sqrt(5)/log(phi) = A002163/A002390, phi = (1+sqrt(5))/2 = A001622, and f(x) = sin(x)/((exp(Pi*x/(2*log(phi)))-1)*(7-2*cos(x))*(3+2*cos(x))). - Gleb Koloskov, Sep 12 2021

More terms from Alois P. Heinz, Aug 06 2017

A050815 Number of positive Fibonacci numbers with n decimal digits.

Original entry on oeis.org

6, 5, 5, 4, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 4, 5, 5, 5, 5

Offset: 1

Patrick De Geest, Oct 15 1999

If n>1 then a(n) = 4 or 5. - Robert Gerbicz, Sep 05 2002

The sequence is almost periodic, see also A072353. - Reinhard Zumkeller, Apr 14 2005

			At length 1 there are 6 such numbers: 1, 1, 2, 3, 5 and 8.

Hans J. H. Tuenter, Table of n, a(n) for n = 1..1001
Andreas Guthmann, Wieviele k-stellige Fibonaccizahlen gibt es?, Archiv der Mathematik, Vol. 59, No. 4 (1992), pp. 334-340.
Ron Knott, The Fibonacci Numbers.
Ron Knott, Fibonacci Numbers and the Golden Section.
Ron Knott, [Number of digits in Fib(i)] : Calculator.
Jan-Christoph Puchta, The Number of k-Digit Fibonacci Numbers, The Fibonacci Quarterly, Vol. 39, No. 4 (2001), pp. 334-335.
Jürgen Spilker, Die Ziffern der Fibonacci-Zahlen, Elemente der Mathematik, Vol. 58 (Birkhäuser 2003), pp. 26-33.
Eric Weisstein's World of Mathematics, Fibonacci Number.
Eric Weisstein's World of Mathematics, Almost Periodic Function.

See A098842 for another version.

Cf. A000045, A001622, A002390, A072354, A105563, A105565, A060384, A097348.

Drop[Last/@Tally[Table[IntegerLength[Fibonacci[n]],{n,505}]],-1] (* Jayanta Basu, Jun 01 2013 *)

Asymptotic mean: lim_{n->oo} (1/n) * Sum_{k=1..n} a(k) = log(10)/log(phi) = 1/A097348 = 4.7849719667... - Amiram Eldar, Jan 12 2022
For n>1, a(n) = 4+[{n*alpha+beta}<{alpha}], where alpha=log(10)/log(phi), beta=log(5)/(2*log(phi)), [X] is the Iverson bracket, {x}=x-floor(x), denotes the fractional part of x, and phi=(1+sqrt(5))/2. - Hans J. H. Tuenter, Jul 20 2025
a(n) = A072354(n+1)-A072354(n), a first-order difference. - Hans J. H. Tuenter, Jul 20 2025

A139342 Decimal expansion of e^(-(1+sqrt(5))/2).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Apr 14 2008

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 13 2019

			0.19828815286220623226788895660486467084208489250129...

Index entries for transcendental numbers.

Cf. A001622, A094214, A104457, A098317, A002390, A139339, A139340, A139341.

evalf(exp(1)^(-(1+sqrt(5))/2),100); # Wesley Ivan Hurt, Feb 09 2017

RealDigits[E^-GoldenRatio,10,120][[1]] (* Harvey P. Dale, Dec 30 2020 *)

exp(-(sqrt(5)+1)/2) \\ Charles R Greathouse IV, May 13 2019

Equals exp(-A001622).

Equals 1/A139341. - Amiram Eldar, Feb 08 2022

Leading zero removed by R. J. Mathar, Feb 05 2009

A202543 Decimal expansion of the number x satisfying e^(x/2) - e^(-x/2) = 1.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Dec 21 2011

See A202537 for a guide to related sequences. The Mathematica program includes a graph.
W. Gawronski et al. in their paper - see ref. below - obtained the asymptotics for the Chebyshev-Stirling numbers. In the algebraic description of the respective "asymptotic coefficients" the number x = 2*log phi, where phi is the golden section, play the central role. - Roman Witula, Feb 02 2015
Also two times the Lévy measure for the continued fraction of the golden section, i.e., A202543/log(2) is the mean number of bits gained from the next convergent of the continued fraction representation. (See also Dan Lascu in links.) - A.H.M. Smeets, Jun 06 2018

			0.9624236501192068949955178268487368462703686...

W. Gawronski, L. L. Littlejohn, and T. Neuschel, Asymptotics of Stirling and Chebyshev-Stirling numbers of the second kind, arXiv:1308.6803 [math.CO], 2013.
W. Gawronski, L. L. Littlejohn, and T. Neuschel, Asymptotics of Stirling and Chebyshev-Stirling numbers of the second kind, Studies in Applied Mathematics by MIT 133 (2014), 1-17.
Dan Lascu, A Gauss-Kuzmintype problem for a family of continued fraction expansions, Journal of Number Theory 133 (2013), 2153-2181.
Index entries for transcendental numbers

Cf. A001622, A002390, A104457, A202537, A202543.

u = 1/2; v = 1/2;
f[x_] := E^(u*x) - E^(-v*x); g[x_] := 1
Plot[{f[x], g[x]}, {x, 0, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .9, 1}, WorkingPrecision -> 110]
RealDigits[r]    (* A202543 *)
RealDigits[ Log[ (3+Sqrt[5])/2], 10, 99] // First (* Jean-François Alcover, Feb 27 2013 *)
RealDigits[ FindRoot[ Exp[x/2] == 1 +  Exp[-x/2] , {x, 0}, WorkingPrecision -> 128][[1, 2]]][[1]] (* Robert G. Wilson v, Jun 13 2018 *)

2*asinh(1/2) \\ Michel Marcus, Jun 24 2018, after A002390

Equals 2*A002390. - A.H.M. Smeets, Jun 06 2018
From Amiram Eldar, Aug 21 2020: (Start)
Equals log(A104457) = log(1 + A001622).
Equals 2*arcsinh(1/2). [corrected by Georg Fischer, Jul 12 2021]
Equals Sum_{k>=0} (-1)^k*binomial(2*k,k)/((2*k+1)*16^k). (End)
Equals Pi*i + Sum_{k>=0} arctanh(phi^(2^k))/2^k, with phi = A001622 and i = sqrt(-1). - Antonio Graciá Llorente, Feb 13 2025

Typo in name fixed by Jean-François Alcover, Feb 27 2013

A107435 Triangle T(n,k), 1<=k<=n, read by rows: T(n,k) = length of Euclidean algorithm starting with n and k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 3, 3, 2, 1, 1, 1, 3, 1, 4, 2, 2, 1, 1, 2, 1, 2, 3, 2, 3, 2, 1, 1, 1, 2, 2, 1, 3, 3, 2, 2, 1, 1, 2, 3, 3, 2, 3, 4, 4, 3, 2, 1, 1, 1, 1, 1, 3, 1, 4, 2, 2, 2, 2, 1, 1, 2, 2, 2, 4, 2, 3, 5, 3, 3, 3, 2, 1, 1, 1, 3, 2, 3, 2, 1, 3, 4, 3, 4, 2, 2, 1

Offset: 1

Philippe Deléham, Jun 09 2005

Consequence of theorem of Gabriel Lamé (1844): the first value of m in this triangle is T(F(m+2), F(m+1)) where F(n) = A000045(n); example: the first 5 is T(F(7), F(6)) = T(13, 8).
From Bernard Schott, May 01 2022: (Start)
Theorem of Gabriel Lamé (1844): The number of divisions necessary to find the greatest common divisor of two natural numbers n > k by means of the Euclidean algorithm is never greater than five times the number of digits of the smaller number k (see link).
This upper bound 5*length(k) is the best possible; the smallest pairs (n, k) for which T(n, k) = 5 * length(k) when length(k) = 1, 2 or 3 are respectively  (F(7), F(6)), (F(12), F(11)) and (F(17), F(16)) where F(n) = A000045(n). This upper bound is not attained when length(k) >= 4. (End)

			13 = 5*2 + 3, 5 = 3*1 + 2, 3 = 2*1 + 1, 2 = 2*1 + 0 = so that T(13,5) = 4.
Triangle begins:
  1
  1 1
  1 2 1
  1 1 2 1
  1 2 3 2 1
  1 1 1 2 2 1
  1 2 2 3 3 2 1
  1 1 3 1 4 2 2 1
  1 2 1 2 3 2 3 2 1
  1 1 2 2 1 3 3 2 2 1
  1 2 3 3 2 3 4 4 3 2 1
  1 1 1 1 3 1 4 2 2 2 2 1
  1 2 2 2 4 2 3 5 3 3 3 2 1
  1 1 3 2 3 2 1 3 4 3 4 2 2 1
  1 2 1 3 1 2 2 3 3 2 4 2 3 2 1
  1 1 2 1 2 3 3 1 4 4 3 2 3 2 2 1
  1 2 3 2 3 3 3 2 3 4 4 4 3 4 3 2 1
  ..............................
Smallest examples with T(n, k) = 5 * length(k) (Theorem of Gabriel Lamé):
13 = 8*1 + 5, 8 = 5*1 + 3, 5 = 3*1 + 2, 3 = 2*1 + 1, 2 = 2*1 + 0 = so that T(13,8) = 5 = 5 * length(8).
144 = 89*1 + 55, 89 = 55*1 + 34, 55 = 34*1 + 21, 34 = 21*1 + 13, 21 = 13*1 + 8, then 5 steps already seen in the previous example, so that T(144,89) = 10 = 5 * length(89).
1597 = 987*1 + 610, 987 = 610*1 + 377, 610 = 377*1 + 233, 377 = 233*1 + 144, 233 = 144*1 + 89, then 10 steps already seen in the previous examples, so that T(1597,987) = 15 = 5 * length(987).

Ross Honsberger, Mathematical Gems II, Dolciani Mathematical Expositions No. 2, Mathematical Association of America, 1976, Chapter 7, A Theorem of Gabriel Lamé, pp. 54-57.
Wacław Sierpiński, Elementary Theory of Numbers, Theorem 12 (Lamé) p. 21, Warsaw, 1964.

Peter Kagey, Table of n, a(n) for n = 1..10000
Gabriel Lamé, Note sur la limite du nombre des divisions dans la recherche du plus grand commun diviseur entre deux nombres entiers, Comptes rendus de l'Académie des sciences XIX, 1844, pages 867-870.
Wikipedia, Euclidean algorithm.
Wikipedia, Gabriel Lamé.

Cf. A000045, A001622, A002390, A034883, A049816, A051010, A055642.

F:= proc(n,k) option remember;
   if n mod k = 0 then 1
   else 1 + procname(k, n mod k)
   fi
end proc:
seq(seq(F(n,k),k=1..n), n=1..15); # Robert Israel, Feb 16 2016

T[n_, k_] := T[n, k] = If[Divisible[n, k], 1, 1 + T[k, Mod[n, k]]];
Table[T[n, k], {n, 1, 15}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 12 2019, after Robert Israel *)

T(n, k) = if ((n % k) == 0, 1, 1 + T(k, n % k)); \\ Michel Marcus, May 02 2022

T(n, k) = A049816(n, k) + 1.
From Robert Israel, Feb 16 2016: (Start)
T(n, k) = 1 if n == 0 (mod k), otherwise T(n, k) = 1 + T(k, (n mod k)).
G.f. G(x,y) of triangle satisfies G(x,y) = x*y/((1-x)*(1-x*y)) - Sum_{k>=1} (x^2*y)^k/(1-x^k) + Sum_{k>=1} G(x^k*y,x). (End)
From Bernard Schott, Apr 29 2022: (Start)
T(F(m+2), F(m+1)) = m where F(n) = A000045(n) (first comment).
T(n, k) <= 5 * length(k) where length(k) = A055642(k).
T(n, k) <= 1 + floor(log(k)/log(phi)) where log(phi) = A002390; the least numbers for which equality stands are when k and n are consecutive Fibonacci numbers. (End)

A202537 Decimal expansion of x satisfying e^x-e^(-2x)=1.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Dec 21 2011

If u>0 and v>0, there is a unique number x satisfying e^(ux)-e^(-vx)=1.  Guide to related sequences, with graphs included in Mathematica programs:
u.... v.... x
1.... 1.... A002390
1.... 2.... A202537
1.... 3.... A202538
2.... 1.... A202539
3.... 1.... A202540
2.... 2.... A202541
3.... 3.... A202542
1/2..1/2... A202543
Suppose that f(x,u,v) is a function of three real variables and that g(u,v) is a function defined implicitly by f(g(u,v),u,v)=0.  We call the graph of z=g(u,v) an implicit surface of f.  For an example related to A202537, take f(x,u,v)=e^(ux)-e^(-vx)-1 and g(u,v) = a nonzero solution x of f(x,u,v)=0.  If there is more than one nonzero solution, care must be taken to ensure that the resulting function g(u,v) is single-valued and continuous.  A portion of an implicit surface is plotted by Program 2 in the Mathematica section.

			0.382245085840035641329358499184857393759416422...

Index entries for transcendental numbers.

Cf. A002390.

(* Program 1:  A202537 *)
u = 1; v = 2;
f[x_] := E^(u*x) - E^(-v*x); g[x_] := 1
Plot[{f[x], g[x]}, {x, -2, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .3, .4}, WorkingPrecision -> 110]
RealDigits[r]  (* A202537 *)
(* Program 2: implicit surface for e^(ux)-e(-vx)=1 *)
f[{x_, u_, v_}] := E^(u*x) - E^(-v*x) - 1;
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, .3}]}, {v, 1, 4}, {u, 2, 20}];
ListPlot3D[Flatten[t, 1]] (* for A202537 *)
First[ RealDigits[ Log[ Root[#^3 - #^2 - 1 & , 1]], 10, 99]] (* Jean-François Alcover, Feb 26 2013 *)

solve(x=0,1,exp(x)-exp(-2*x)-1) \\ Charles R Greathouse IV, Feb 26 2013

log(polrootsreal(x^3-x^2-1)[1]) \\ Charles R Greathouse IV, Feb 07 2025

Digits from a(90) on corrected by Jean-François Alcover, Feb 26 2013

cp's OEIS Frontend

A139341 Decimal expansion of e^((1+sqrt(5))/2).

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A153386 Decimal expansion of Sum_{n>=1} 1/Fibonacci(2*n).

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A263401 Expansion of Product_{k>=1} (1 + x^k - x^(2*k)).

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A139340 Decimal expansion of the cube root of the golden ratio. That is, the decimal expansion of ((1+sqrt(5))/2)^(1/3).

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A290565 Decimal expansion of sum of reciprocal golden rectangle numbers.

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A050815 Number of positive Fibonacci numbers with n decimal digits.

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A139342 Decimal expansion of e^(-(1+sqrt(5))/2).

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Maple