A002391 -id:A002391 - cp's OEIS frontend

A153459 Decimal expansion of log_3 (6).

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

Offset: 1

N. J. A. Sloane, Oct 30 2009

Equals the Hausdorff dimension of Pascal's triangle modulo 3 (A083093). In general, the dimension of Pascal's triangle modulo a prime p is log(p*(p+1)/2) / log(p) (see Reiter link, theorem 2 page 117). - Bernard Schott, Dec 01 2022

			1.6309297535714574370995271143427608542995856401318804278706...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000.
A. M. Reiter, Determining the dimension of fractals generated by Pascal's triangle, Fibonacci Quarterly, 31(2), 1993, pp. 112-120.
Wikipedia, List of fractals by Hausdorff dimension (see Pascal triangle modulo 3).
Index entries for transcendental numbers

cf. A002391, A016629, A083093, A102525.

evalf(log(6)/log(3),80); # Bernard Schott, Dec 01 2022

RealDigits[Log[3, 6], 10, 120][[1]] (* Vincenzo Librandi, Aug 29 2013 *)

Equals A016629 / A002391 = 1 + A102525. - Bernard Schott, Dec 01 2022

A254620 a(n) = 9^n(2n + 1)!/n!.

Original entry on oeis.org

1, 54, 4860, 612360, 99202320, 19642059360, 4596241890240, 1240985310364800, 379741504971628800, 129871594700297049600, 49091462796712284748800, 20323865597838885886003200, 9145739519027498648701440000, 4444829406247364343268899840000

Offset: 0

Peter Bala, Feb 03 2015

Cf. A002162, A002391, A034910, A073000, A105531, A254381, A254619.

Maple
```
seq(9^n*(2*n + 1)!/n!, n = 0..14);
```

Table[9^n (2n+1)!/n!,{n,0,20}] (* Harvey P. Dale, Aug 13 2019 *)

E.g.f.: 1/(1 - 36*x)^(3/2) = 1 + 54*x + 4860*x^2/2! + 612360*x^3/3! + ....
Recurrence equation: a(n) = 18*(2*n + 1)*a(n-1) with a(0) = 1.
2nd order recurrence equation: a(n) = (40*n + 16)*a(n-1) - 36*(2*n - 1)^2*a(n-2) with a(0) = 1, a(1) = 54.
Define a sequence b(n) := a(n)*sum {k = 0..n} 1/((2*k + 1)*9^k) beginning [1, 56, 5052, 636672, 103142544, 20422253952, 4778808090048, ...]. It is not difficult to check that b(n) also satisfies the previous 2nd order recurrence equation (and so is an integer sequence). Using this observation we obtain the continued fraction expansion log(2) = 2/3*Sum {k >= 0} 1/((2*k + 1)*9^k) = 2/3*(1 + 2/(54 - 36*3^2/(96 - 36*5^2/(136 - ... - 36*(2*n - 1)^2/((40*n + 16) - ... ))))).
Alternative 2nd order recurrence equation: a(n) = (32*n + 20)*a(n-1) + 36*(2*n - 1)^2*a(n-2) with a(0) = 1, a(1) = 54.
Define now a sequence c(n) := a(n)*sum {k = 0..n} (-1)^k/((2*k + 1)*9^k) beginning [1, 52, 4692, 591072, 95755344, 18959527872, 4436530187328, ...], which, along with a(n), satisfies the alternative 2nd order recurrence equation. From this observation we find the continued fraction expansion arctan(1/3) = 1/3*Sum {k >= 0} (-1)^k/((2*k + 1)*9^k) = 1/3*(1 - 2/(54 + 36*3^2/(84 + 36*5^2/(116 + ... + 36*(2*n - 1)^2/((32*n + 20) + ... ))))). Cf. A254381 and A254619.

A325752 First term of n-th difference sequence of (floor(r*k)), r = log(3), k >= 0.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -11, 66, -286, 1001, -3003, 8008, -19448, 43758, -92378, 184757, -352737, 646877, -1145837, 1971882, -3321890, 5541965, -9324315, 16231215, -30045015, 60090031, -129024511, 290305576, -665732936, 1523150156, -3431847188

Offset: 1

Clark Kimberling, Jun 07 2019

Clark Kimberling, Table of n, a(n) for n = 1..200
Sean A. Irvine, Java program (github)

Cf. A002391, A059543.

Table[First[Differences[Table[Floor[n*Log[3]], {n, 0, 50}], n]], {n, 1, 50}]

A382778 Decimal expansion of 6log(3)/(3log(3) - 3).

Original entry on oeis.org

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

Offset: 2

Stefano Spezia, May 11 2025

Upper bound for the irrationality measure of 3-adic analog of zeta(3) (see Lai et al., 2025 at p. 3).

			22.2814479514943215603962067415858532334689249078...

Li Lai, Johannes Sprang, and Wadim Zudilin, A note on the irrationality of zeta_2(5), arXiv:2505.05005 [math.NT], 2025.
Eric Weisstein's World of Mathematics, Irrationality Measure.
Wikipedia, Irrationality measure.

Cf. A002391, A016650, A382497, A383822, A383824.

RealDigits[6Log[3]/(3Log[3]-3),10,100][[1]]

A386736 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} {1/(x+y+z)}^2 dx dy dz, where {} denotes fractional part.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Aug 01 2025

			0.3885477223540393286722981299552364311798731004354002...

Ovidiu Furdui, Multiple Fractional Part Integrals and Euler's Constant, Miskolc Mathematical Notes, Vol. 17, No. 1 (2016), pp. 255-266.

Cf. A002117, A002162, A002391, A013661, A386734, A386735, A386737.

RealDigits[6*Log[2] - 3*Log[3] - (Zeta[2] + Zeta[3])/6, 10, 120][[1]]

6*log(2) - 3*log(3) -(zeta(2) + zeta(3))/6

Equals 6*log(2) - 3*log(3) -(zeta(2) + zeta(3))/6.

A386737 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} {1/(x+y+z)}^3 dx dy dz, where {} denotes fractional part.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Aug 01 2025

			0.27606787380471479458379157265271548892384688537591...

Ovidiu Furdui, Multiple Fractional Part Integrals and Euler's Constant, Miskolc Mathematical Notes, Vol. 17, No. 1 (2016), pp. 255-266.

Cf. A001620 (gamma), A002117, A002162, A002391, A013661, A386734, A386736.

RealDigits[Log[3]/2 - 3*Log[2]/2 + 5/3 - EulerGamma/2 - Zeta[2]/4 - Zeta[3]/6, 10, 120][[1]]

log(3)/2 - 3*log(2)/2 + 5/3 - Euler/2 - zeta(2)/4 - zeta(3)/6

Equals log(3)/2 - 3*log(2)/2 + 5/3 - gamma/2 - zeta(2)/4 - zeta(3)/6.

A059181 Engel expansion of log(3) = 1.09861... .

Original entry on oeis.org

1, 11, 12, 60, 108, 139, 176, 1228, 1356, 3166, 14807, 81596, 116387, 1367315, 4408018, 11560054, 15330821, 448349063, 574897948, 613663772, 636869505, 999446566, 3786082000, 6911150147, 19780703830, 25945009665, 33227233233, 103529753474, 118556676397

Offset: 1

Mitch Harris

Cf. A006784 for definition of Engel expansion.

F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191.

G. C. Greubel, Table of n, a(n) for n = 1..1000
F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
P. Erdős and Jeffrey Shallit, New bounds on the length of finite Pierce and Engel series, Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
Index entries for sequences related to Engel expansions

Cf. A002391.

EngelExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@
NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]/1} &, {Ceiling[1/(A - Floor[A])], (A - Floor[A])/1}, n - 1]];
EngelExp[N[Log[3], 7!], 100] (* Modified by G. C. Greubel, Dec 27 2016 *)

A068433 Expansion of log(3) in base 2.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1

Offset: 1

Benoit Cloitre, Mar 09 2002

			The decimal expansion is log(3) = 1.098612288668109...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A002391.

function binaryExp(n) {
x = new Array();
for (i = 0; i < 50; i++) if (n - Math.pow(2, 1 - i) >= 0) {n -= Math.pow(2, 1 - i); x[i] = 1;} else x[i] = 0;
return x;
}
document.write(Math.log(3) + "
");
document.write(binaryExp(Math.log(3))); // Jon Perry, Nov 18 2012

RealDigits[Log[3], 2, 108][[1]] (* Alonso del Arte, Feb 17 2013 *)

concat(binary(log(3))) \\ Michel Marcus, Dec 14 2017

A157024 a(0)=1, a(n) = (3n-1)3n(3n+1)/2 for n>0.

Original entry on oeis.org

1, 12, 105, 360, 858, 1680, 2907, 4620, 6900, 9828, 13485, 17952, 23310, 29640, 37023, 45540, 55272, 66300, 78705, 92568, 107970, 124992, 143715, 164220, 186588, 210900, 237237, 265680, 296310, 329208, 364455, 402132, 442320, 485100, 530553, 578760, 629802

Offset: 0

Jaume Oliver Lafont, Feb 21 2009

nonn

Cf. A002391, A027480, A058962, A097321, A154920.

nxt[{n_,a_}]:={n+1,((3n)(3n-1)(3n+1))/2}; NestList[nxt,{0,1},40][[All,2]]/.(0->Nothing) (* Harvey P. Dale, Sep 24 2016 *)

Sum_{n>=0} 1/a(n) = log(3).
G.f.: (1+8x+63x^2+8x^3+x^4)/(1-x)^4.
a(n) = A027480(3n-1), n>0. - R. J. Mathar, Apr 07 2009
Sum_{n>=0} (-1)^n/a(n) = 4*log(2)/3. - Amiram Eldar, Feb 27 2022

A293812 Decimal expansion of log(3)/log(1 + sqrt(2)).

Original entry on oeis.org

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

Offset: 1

Arkadiusz Wesolowski, Oct 16 2017

Fractal dimension of the frontier of the Fibonacci word fractal.

			1.24647743572981584189100424874815183996105530003376417796845193354456...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Alexis Monnerot-Dumaine, The Fibonacci Word Fractal, HAL-00367972, February 2009, section 5.2.
Alexis Monnerot-Dumaine, The Fibonacci word fractal [Cached copy, with permission]
Index entries for transcendental numbers

Equals A002391 / A091648.

SetDefaultRealField(RealField(105)); n:=Log(1+Sqrt(2),3); Reverse(Intseq(Floor(10^104*n)));

evalf(log(3)/log(1+sqrt(2)),110); # Muniru A Asiru, Oct 11 2018

RealDigits[Log[3]/Log[1 + Sqrt[2]], 10, 100][[1]] (* G. C. Greubel, Oct 10 2018 *)

PARI
```
log(3)/log(1+sqrt(2))
```

cp's OEIS Frontend

A153459 Decimal expansion of log_3 (6).

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A254620 a(n) = 9^n*(2*n + 1)!/n!.

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A325752 First term of n-th difference sequence of (floor(r*k)), r = log(3), k >= 0.

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A382778 Decimal expansion of 6*log(3)/(3*log(3) - 3).

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A386736 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} {1/(x+y+z)}^2 dx dy dz, where {} denotes fractional part.

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A386737 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} {1/(x+y+z)}^3 dx dy dz, where {} denotes fractional part.

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A059181 Engel expansion of log(3) = 1.09861... .

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A068433 Expansion of log(3) in base 2.

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A254620 a(n) = 9^n(2n + 1)!/n!.

A382778 Decimal expansion of 6log(3)/(3log(3) - 3).

A157024 a(0)=1, a(n) = (3n-1)3n(3n+1)/2 for n>0.