A002390 -id:A002390 - cp's OEIS frontend

A129269 Decimal expansion of arcsinh(1/5).

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

Offset: 0

N. J. A. Sloane, Jul 27 2008

			0.19869011034924140647463691595020696822130879422445377302126...

Index entries for transcendental numbers

Decimal expansion of arcsinh(1/k): A091648 (k=1), A002390 (k=2), A129187(k=3), A129200 (k=4).

RealDigits[ArcSinh[1/5], 10, 105][[1]] (* Amiram Eldar, May 02 2023 *)

asinh(.2) \\ Charles R Greathouse IV, Nov 21 2024

A160509 Decimal expansion of 1/log(phi).

Original entry on oeis.org

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

Offset: 1

Hagen von Eitzen, May 16 2009

			2.07808692123502753760132260611779576774219226778328...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.18.2, p. 159.
D. E. Knuth, The Art of Computer Programming, Vol 1: Fundamental Algorithms, Addison-Wesley, 1968, Appendix B, Table 1.

J. D. Cloud, Problem E 1636, The American Mathematical Monthly, Vol. 70, No. 9 (1963), p. 1005; Number of Fibonacci Numbers Not Exceeding N, Solution to Problem E 1636 by William D. Jackson, ibid., Vol. 71, No. 7 (1964), p. 798.
H. W. Gould, Non-Fibonacci Numbers, The Fibonacci Quarterly, Vol. 3, No. 3 (1965), pp. 177-183.
Douglas Lind, Problem H-74, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 3, No. 4 (1965), p. 300; A Better Problem Solution, Solution to Problem H-74, ibid., Vol. 4, No. 1 (1966), pp. 58-59.
Index entries for transcendental numbers.

Cf. A001622, A002390, A072649.

RealDigits[1/Log@GoldenRatio, 10, 111][[1]] (* Robert G. Wilson v, May 29 2009 *)

1/asinh(1/2) \\ Charles R Greathouse IV, Jan 04 2016

From Amiram Eldar, Feb 05 2022: (Start)
Equals 1/A002390.
Equals lim_{n->oo} A072649(n)/log(n) (Cloud, 1963). (End)

More terms from Robert G. Wilson v, May 29 2009

A174607 Decimal expansion of Pi^2/(6*log(phi)) where phi=(1+sqrt(5))/2.

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Mar 23 2010

			3.41831597061124385292763138...

Brigitte Vallée, Digits and continuants in Euclidean algorithms. Ergodic vs tauberian theorems, Journal de théorie des nombres de Bordeaux 12 (2000), 519-558.

Cf. A001622, A002390, A013661.

RealDigits[Pi^2/(6*Log[GoldenRatio]), 10, 100][[1]] (* Amiram Eldar, May 24 2023 *)

Pi^2/6/log((1+sqrt(5))/2) \\ Michel Marcus, Apr 04 2014

A234368 Floor(AGM(1, Fibonacci(n))), where AGM denotes the arithmetic-geometric mean.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 5, 7, 10, 16, 23, 35, 53, 80, 122, 187, 286, 439, 675, 1041, 1608, 2491, 3864, 6004, 9344, 14563, 22729, 35517, 55566, 87026, 136440, 214117, 336322, 528725, 831868, 1309817, 2063861, 3254227, 5134497, 8106188, 12805344, 20239959

Offset: 1

Alex Ratushnyak, Dec 25 2013

nonn

			a(9) = floor(AGM(1, Fibonacci(9))) = floor(AGM(1, 34)) = 10.

Cf. A000040, A127758.

Table[Floor[ArithmeticGeometricMean[1, Fibonacci[n]]], {n, 1, 50}] (* Vaclav Kotesovec, May 09 2016 *)

a(n) ~ Pi * phi^n / (2 * sqrt(5) * log(phi) * n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio, log(phi) = A002390. - Vaclav Kotesovec, May 09 2016

A318057 a(n) is the number of binary places to which n-th convergent of continued fraction expansion of the golden section matches the correct value.

Original entry on oeis.org

0, -2, 3, 2, 5, 2, 6, 9, 10, 9, 13, 12, 15, 16, 19, 16, 20, 22, 24, 25, 27, 29, 28, 30, 33, 32, 36, 32, 38, 32, 41, 42, 44, 45, 46, 47, 50, 48, 52, 54, 53, 56, 53, 58, 59, 60, 64, 62, 66, 62, 67, 69, 71, 73, 75, 74, 77, 78, 80, 82, 81, 84, 81, 87, 81, 88, 90

Offset: 1

A.H.M. Smeets, Aug 14 2018

The correct binary value of the golden section is given in A068432; the continued fraction terms of the golden section is given in A000012.
For the number of correct decimal digits of the golden section see A318058.
The denominator of the k-th convergent obtained from a continued fraction tend to k*A001622; the error between the k-th convergent and the constant itself tends to 1/(2*k*A001622), or in binary digits 2*k*log(A001622)/log(2) bits after the binary point.
The sequence for quaternary digits is obtained by floor(a(n)/2), the sequence for octal digits is obtained by floor(a(n)/3), and the sequence for hexadecimal digits is obtained by floor(a(n)/4).

			   n   convergent         binary expansion       a(n)
  ==  =============  ==========================  ====
   1    1 / 1          1.0                         0
   2    2 / 1         10.0                        -2
   3    3 / 2          1.1000                      3
   4    5 / 3          1.101                       2
   5    8 / 5          1.100110                    5
   6   13 / 8          1.101                       2
   7   21 / 13         1.1001110                   6
   8   34 / 21         1.1001111001                9
   9   55 / 34         1.10011110000              10
  10   89 / 55         1.1001111001                9
  oo  lim = A068432    1.1001111000110111011110   --

Cf. A000012, A001622, A002390, A068432, A202543, A242208, A318058.

p, q, i, base = 1, 1, 0, 2
while i < 20200:
    p, q, i = p+q, p, i+1
a0, p, q = p//q, q, p
i, p, dd = 0, p*base, [0]
while i < 30000:
    d, p, i = p//q, (p%q)*base, i+1
    dd = dd+[d]
n, pn, qn = 0, 1, 0
while n < 20000:
    n, pn, qn = n+1, pn+qn, pn
    if pn//qn != a0:
        print(n, "- manual!")
    else:
        i, p, q, di = 0, (pn%qn)*base, qn, 0
        while di == dd[i]:
            i, di, p = i+1, p//q, (p%q)*base
        print(n, i-1)

Lim {n -> oo} a(n)/n = 2*log(A001622)/log(2) = 2*A002390/log(2) = A202543/log(2) = 2*A242208.

A318058 a(n) is the number of decimal places to which the n-th convergent of the continued fraction expansion of the golden section matches the correct value.

Original entry on oeis.org

0, -1, 0, 1, 1, 1, 2, 2, 2, 3, 2, 4, 4, 5, 5, 5, 6, 5, 7, 7, 8, 7, 9, 9, 9, 10, 10, 10, 11, 10, 12, 12, 13, 12, 13, 14, 15, 14, 15, 16, 16, 16, 17, 17, 17, 18, 18, 19, 18, 20, 20, 21, 21, 21, 22, 22, 23, 23, 24, 23, 24, 25, 25, 26, 26, 27, 27, 27, 28, 28, 29, 29, 29, 30, 30

Offset: 1

A.H.M. Smeets, Aug 14 2018

The correct decimal value of the golden section is given in A001622; the continued fraction terms of the golden section is given in A000012.
For the number of correct decimal digits of the golden section see A318057.
The denominator of the k-th convergent obtained from a continued fraction tend to k*A001622; the error between the k-th convergent and the constant itself tends to 1/(2*k*A001622), or in binary digits 2*k*log(A001622)/log(2) bits after the binary point.

			   n   convergent         decimal expansion     a(n)
  ==  =============  =========================  ====
   1    1 / 1         1.0                         0
   2    2 / 1         2.0                        -1
   3    3 / 2         1.5                         0
   4    5 / 3         1.66                        1
   5    8 / 5         1.60                        1
   6   13 / 8         1.62                        1
   7   21 / 13        1.615                       2
   8   34 / 21        1.619                       2
   9   55 / 34        1.617                       2
  10   89 / 55        1.6181                      3
  oo  lim = A001622   1.6180339887498948482      --

Cf. A000012, A001622, A002390, A097348, A202543, A318057.

p, q, i, base = 1, 1, 0, 10
while i < 20200:
    p, q, i = p+q, p, i+1
a0, p, q = p//q, q, p
i, p, dd = 0, p*base, [0]
while i < 30000:
    d, p, i = p//q, (p%q)*base, i+1
    dd = dd+[d]
n, pn, qn = 0, 1, 0
while n < 20000:
    n, pn, qn = n+1, pn+qn, pn
    if pn//qn != a0:
        print(n, "- manual!")
    else:
        i, p, q, di = 0, (pn%qn)*base, qn, 0
        while di == dd[i]:
            i, di, p = i+1, p//q, (p%q)*base
        print(n, i-1)

Limit_{n -> oo} a(n)/n = 2*log(A001622)/log(10) = 2*A002390/log(10) = A202543/log(10) = 2*A097348.

A338303 Decimal expansion of Sum_{k>=0} 1/(L(2*k) + 2), where L(k) is the k-th Lucas number (A000032).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Oct 21 2020

Backstrom (1981) found that the sum is approximately equal to 1/8 + 1/(4*log(phi)), where phi is the golden ratio (A001622). The difference is less than 1/10^7. He called this difference "a tantalizing problem".
Almkvist (1986) added a term to Backstrom's formula to get an even better approximation which differs from the exact value by less than 1/10^33: 1/8 + 1/(4*log(phi)) + Pi^2/(log(phi)^2 * (exp(Pi^2/log(phi)) - 2)). He found an exact formula from a quotient of two Jacobi theta functions (see the FORMULA section), and showed that both approximations are just the first terms in a rapidly converging series.
Since exp(-Pi^2/log(phi)) = 1.23...*10^(-9) is small, the convergence is rapid: the number of terms needed in each of the two series in the formula to get 10^2, 10^3 and 10^4 decimal digits are merely 3, 10 and 32, respectively.
Andre-Jeannin (1991) noted that if the summand 2 that is added to the Lucas numbers is replaced with sqrt(5), then the sum is Sum_{k>=0} 1/(L(2*k) + sqrt(5)) = 1/phi (A094214).

			0.64452178306727444209927311903801690292890812387799...

Jonathan M. Borwein and Peter B. Borwein, Pi and the AGM, Wiley, 1987, p. 99.

Gert Almkvist, A solution to a tantalizing problem, The Fibonacci Quarterly, Vol. 24, No. 4 (1986), pp. 316-322.
Richard Andre-Jeannin, Summation of certain reciprocal series related to Fibonacci and Lucas numbers, The Fibonacci Quarterly, Vol. 29, No. 3 (1991), pp. 200-204.
Robert P. Backstrom, On reciprocal series related to Fibonacci numbers with subscripts in arithmetic progression, The Fibonacci Quarterly, Vol. 19, No. 1 (1981), pp. 14-21. See section 6, pp. 19-20.
Daniel Duverney and Iekata Shiokawa, On series involving Fibonacci and Lucas numbers I, AIP Conference Proceedings, Vol. 976, No. 1 (2008), pp. 62-76, alternative link.

Cf. A000032, A001622, A002390, A094214, A160509, A240926.

With[{lg = Log[GoldenRatio], kmax = 3}, sum[m_, k_] := (-1)^k*k^m*Exp[-Pi^2*k^2/lg]; RealDigits[1/8 + ( 1/(4*lg))*(1 - (4*Pi^2/lg)*(Sum[sum[2, k], {k, 1, kmax}]/(1 + 2*Sum[sum[0, k], {k, 1, kmax}]))), 10, 100][[1]]]

Equals Sum_{k>=0} 1/A240926(k).
Equals 1/4 + Sum_{k>=1} q^(2*k)/(1 + q^(2*k))^2, where q = 1/phi.
Equals 1/8 + (1/(4*log(phi))) * (1 - (4*Pi^2/log(phi)) * (S(2)/(1 + 2*S(0)))), where S(m) = Sum_{k>=1} (-1)^k * k^m * exp(-Pi^2*k^2/log(phi)) (Almkvist, 1986).

A344475 Decimal expansion of the value of the Dickman function at phi + 1 = phi^2 = (3 + sqrt(5))/2 (A104457).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, May 20 2021

			0.10464776377316485385416972771819339482414269115729...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 286.

Pieter Moree, A special value of Dickman's function, Math. Student, Vol. 64 (1995), pp. 47-50; entire volume.
Pieter Moree, Nicolaas Govert de Bruijn, the enchanter of friable integers, Indagationes Mathematicae, Vol. 24, No. 4 (2013), pp. 774-801.
Eric Weisstein's World of Mathematics, Dickman Function.
Wikipedia, Dickman function.

Cf. A000796, A001622, A002390, A013661, A104457, A344476.

Cf. A175475, A244009 (value at 1/2), A245238, A309638.

RealDigits[1 - 2*Log[GoldenRatio] + Log[GoldenRatio]^2 - Pi^2/60, 10, 100][[1]]

my(phi = quadgen(5)); 1 - 2*log(phi) + log(phi)^2 - Pi^2/60 \\ Amiram Eldar, Jan 09 2025

Equals 1 - 2*log(phi) + log(phi)^2 - Pi^2/60 (Moree, 1995).

More terms from Amiram Eldar, Jan 09 2025

A344476 Decimal expansion of the value of the Buchstab function at phi + 2 = (5 + sqrt(5))/2 (A296184).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, May 20 2021

			0.56093863969277163346004116368055699296113179704964...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 286.

A. A. Bukhstab, Asymptotic estimates of a general number-theoretic function (in Russian), Matematicheskii Sbornik, Vol. 44, No. 6 (1937) pp. 1239-1246.
Pieter Moree, A special value of Dickman's function, Math. Student, Vol. 64 (1995), pp. 47-50; entire volume.
Eric Weisstein's World of Mathematics, Buchstab Function.
Wikipedia, Buchstab function.

Cf. A000796, A001622, A002390, A013661, A296184, A344475.

RealDigits[(1 + 2*Log[GoldenRatio] + Log[GoldenRatio]^2 - Pi^2/60)/(GoldenRatio + 2), 10, 100][[1]]

my(phi = quadgen(5)); (1 + 2*log(phi) + log(phi)^2 - Pi^2/60)/(phi+2) \\ Amiram Eldar, Jan 09 2025

Equals (1 + 2*log(phi) + log(phi)^2 - Pi^2/60)/(phi+2) (Moree, 1995).

More terms from Amiram Eldar, Jan 09 2025

A351789 Decimal expansion of Sum_{k>=1} AH(k)*F(k)/2^k, where AH(k) = A058313(k)/A058312(k) is the k-th alternating harmonic number and F(k) = A000045(k) is the k-th Fibonacci number.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Feb 19 2022

			1.51437037420622187243459478916165077964831313316887...

Seán M. Stewart, Problem H-893, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 60, No. 1 (2022), p. 91.

Cf. A000045, A001622, A002390, A058312, A058313, A349850, A351794.

RealDigits[Log[5/4] + 6*Log[GoldenRatio]/Sqrt[5], 10, 100][[1]]

Equals log(5/4) + 6*log(phi)/sqrt(5), where phi is the golden ratio (A001622) (Stewart, 2022).

cp's OEIS Frontend

A129269 Decimal expansion of arcsinh(1/5).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A160509 Decimal expansion of 1/log(phi).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A174607 Decimal expansion of Pi^2/(6*log(phi)) where phi=(1+sqrt(5))/2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A234368 Floor(AGM(1, Fibonacci(n))), where AGM denotes the arithmetic-geometric mean.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A318057 a(n) is the number of binary places to which n-th convergent of continued fraction expansion of the golden section matches the correct value.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Python

Formula

A318058 a(n) is the number of decimal places to which the n-th convergent of the continued fraction expansion of the golden section matches the correct value.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Python

Formula

A338303 Decimal expansion of Sum_{k>=0} 1/(L(2*k) + 2), where L(k) is the k-th Lucas number (A000032).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A344475 Decimal expansion of the value of the Dickman function at phi + 1 = phi^2 = (3 + sqrt(5))/2 (A104457).

Views

Author

Keywords

Examples