cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A097348 Decimal expansion of arccsch(2)/log(10).

Original entry on oeis.org

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

Views

Author

Ed Pegg Jr, Aug 06 2004

Keywords

Comments

First n terms give number of digits of Fibonacci(10^n), except that it can be off by 1. This is a highly compressed sequence. As a result, it can be off by one. The uncompressed version goes like this: 2, 21, 209, 2090, 20899, 208988, 2089877, 20898764, 208987640, 2089876403, ... (see A068070). Fibonacci(10) = 55 has 2 digits, Fibonacci(100) = 354224848179261915075 has 21 digits and so on.
Considering the very good approximation F(n) = 5^(-1/2)*phi^n, the number of digits of F(10^n) is given by floor(log_10(F(10^n))) = floor(-(1/2)*log_10(5) + 10^n*log_10(phi)). Similarly L(n) tends to phi^n, so the number of digits of L(10^n) is given by floor(10^n*log_10(phi)). Both numbers can differ at most by 1. F(n) and L(n) denote the Fibonacci and Lucas numbers, resp. - Christoph Pacher (christoph.pacher(AT)arcs.ac.at), Nov 22 2006

Examples

			0.20898764024997873376...
Fibonacci(10^9)  has 208987640 decimal digits;
Fibonacci(10^21) has 208987640249978733769 decimal digits;
Fibonacci(10^27) has 208987640249978733769272089 decimal digits.

Links

Crossrefs

Cf. A000045, A001622, A068070, A114469.

Programs

  • Maple
    phi := (1+sqrt(5))/2 ; evalf( log(phi)/log(10)) ; # R. J. Mathar, Oct 17 2012
  • Mathematica
    FibonacciDigits[n_] := Ceiling[(2*n*ArcCsch[2] - Log[5])/Log[100]]
RealDigits[ArcCsch[2]/Log[10], 10, 105][[1]] (* Vaclav Kotesovec, Aug 09 2015 *)
  • PARI
    solve(x=.1,1,sinh(x)-.5)/log(10) \\ Charles R Greathouse IV, Aug 04 2020
  • PARI
    log((1+sqrt(5))/2)/log(10) \\ Charles R Greathouse IV, Aug 04 2020

Formula

Equals log_10(phi) where phi = (1+sqrt(5))/2. - Jaroslav Krizek, Dec 23 2013

Extensions

Offset corrected by Lee A. Newberg, Oct 13 2022

A114469 Number of decimal digits in Lucas(10^n).

Original entry on oeis.org

1, 3, 21, 209, 2090, 20899, 208988, 2089877, 20898765, 208987641, 2089876403, 20898764025, 208987640250, 2089876402500, 20898764024998, 208987640249979, 2089876402499788, 20898764024997874, 208987640249978734, 2089876402499787338, 20898764024997873377, 208987640249978733770, 2089876402499787337693, 20898764024997873376928, 208987640249978733769273, 2089876402499787337692721
Offset: 0

Views

Author

Eric W. Weisstein, Nov 30 2005

Keywords

Comments

From Hans J. H. Tuenter, Jul 15 2025: (Start)
This sequence can be constructed by taking the first n digits of the decimal expansion of log_10(phi) = A097348 and adding 1. For example,
a(0) = 1,
a(1) = 2+1 = 3,
a(2) = 20+1 = 21,
a(3) = 208+1 = 209,
a(4) = 2089+1 = 2090,
a(5) = 20898+1 = 20899.
Alternatively, a(n)-1 is the first n digits of A097348. (End)
The individual digits of log_10(phi), and thus sequence A097348, can be extracted using d(n) = a(n)-10*a(n-1)+9. - Hans J. H. Tuenter, Jul 25 2025

Links

Crossrefs

Cf. A068070. - R. J. Mathar, Dec 13 2008
Cf. A000032, A001622, A055642, A097348.

Programs

Formula

Limit_{n->oo} a(n)/10^n = A097348. - Amiram Eldar, Apr 14 2022
a(n) = 1+floor(10^n*log_10(phi)), where phi = (1+sqrt(5))/2, the golden ratio. - Hans J. H. Tuenter, Jul 13 2025.
From Hans J. H. Tuenter, Jul 25 2025: (Start)
a(n) = 1 + Sum_{i=0..n} d(i)*10^(n-i),
a(n) = 10*a(n-1)-9+d(n),
where d(i) = A097348(i-1) is the coefficient of 10^(-i) in the decimal expansion of log_10(phi). (End)

Extensions

a(8)-a(10) from Amiram Eldar, Apr 14 2022
a(11)-a(25) from Hans J. H. Tuenter, Jul 15 2025

A385248 Number of digits in the decimal expansion of Fibonacci(2^n).

Original entry on oeis.org

1, 1, 1, 2, 3, 7, 14, 27, 54, 107, 214, 428, 856, 1712, 3424, 6848, 13696, 27393, 54785, 109570, 219140, 438279, 876558, 1753116, 3506231, 7012462, 14024923, 28049846, 56099693, 112199385, 224398770, 448797540, 897595080, 1795190160, 3590380321, 7180760641
Offset: 0

Views

Author

Juande Santander-Vela, Jul 28 2025

Keywords

Comments

Binet's formula is Fibonacci(k) = (phi^k - psi^k)/sqrt(5), with phi being the golden ratio (1 + sqrt(5))/2, and psi = (1 - sqrt(5))/2. For even values of k, Fibonacci(k) = floor((phi^k)/sqrt(5)) since psi^(2*k)/sqrt(5) < 0.17^k for all k > 0, and from which the formula below.

Links

Crossrefs

Cf. A000079, A000045, A001622, A055642, A058635, A060384, A068070.

Programs

  • Maple
    a:= n-> `if`(n=0, 1, floor(2^n*log[10]((1+sqrt(5))/2)-log[10](5)/2)+1):
seq(a(n), n=0..35);  # Alois P. Heinz, Jul 30 2025
  • Mathematica
    a[n_] := IntegerLength[Fibonacci[2^n]]; Array[a, 30, 0] (* Amiram Eldar, Jul 30 2025 *)
A385248[n_] := If[n == 0, 1, Floor[2^n*Log10[GoldenRatio] - Log10[5]/2] + 1];
Array[A385248, 50, 0] (* Paolo Xausa, Aug 07 2025 *)
  • PARI
    a(n) = #Str(fibonacci(2^n)); \\ Michel Marcus, Jul 30 2025
  • Python
    from sympy import Rational, log, sqrt # uses symbolic computation
phi = (1+sqrt(5))/2
def a(n): return 1 if n==0 or n==1 else int(2**n *log(phi)/log(10)-Rational(1,2)*log(5)/log(10))+1

Formula

a(n) = A055642(A058635(n)).
a(n) = A060384(A000079(n)).
a(n) = floor(2^n * log10(phi) - (1/2) * log10(5)) + 1, for n >= 1.
Limit_{n->oo} a(n+1)/a(n) = 2.

Extensions

More terms from Michel Marcus, Jul 30 2025
a(29)-a(35) from Amiram Eldar, Jul 30 2025

A096233 Number of digits of the 10^n-th tribonacci number (A000073).

Original entry on oeis.org

2, 26, 264, 2646, 26464, 264649, 2646494, 26464944, 264649443, 2646494434, 26464944348, 264649443484, 2646494434842, 26464944348425, 264649443484250, 2646494434842508, 26464944348425087
Offset: 0

Views

Author

Michael Taktikos, Aug 09 2004

Keywords

Comments

a(n)/10^n tends towards log[10](rho) = .26464944348425087191..., where rho is the real root of x^3-x^2-x-1 = 0. - Vladeta Jovovic, Sep 01 2004

Crossrefs

Cf. A068070.

Programs

  • Mathematica
    rho=1/3*((19+3*Sqrt[33])^(1/3)+(19-3*Sqrt[33])^(1/3)+1); triboappr[n_]:=N[(rho-1)/(4rho-6)*rho^(n-3), 3000]; Table[MantissaExponent[triboappr[10^i]][[2]], {i, 1, 7}]
Showing 1-4 of 4 results.

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