A083679 -id:A083679 - cp's OEIS frontend

A036290 a(n) = n*3^n.

0, 3, 18, 81, 324, 1215, 4374, 15309, 52488, 177147, 590490, 1948617, 6377292, 20726199, 66961566, 215233605, 688747536, 2195382771, 6973568802, 22082967873, 69735688020, 219667417263, 690383311398, 2165293113021, 6778308875544, 21182215236075, 66088511536554, 205891132094649

Offset: 0

N. J. A. Sloane

If X_1,X_2,...,X_n is a partition of a 3n-set X into 3-blocks then, for n > 0, a(n) is equal to the number of (n+1)-subsets of X intersecting each X_i (i=1,2,...,n). - Milan Janjic, Jul 21 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Milan Janjic, Two Enumerative Functions.
László Németh, The trinomial transform triangle, Journal of Integer Sequences, Vol. 21 (2018), Article 18.7.3. Also arXiv:1807.07109 [math.NT], 2018.
Index entries for linear recurrences with constant coefficients, signature (6,-9).

Cf. A000244, A006234, A016578, A027471, A083679, A289399 (partial sums).

[n*3^n: n in [0..30]]; // Vincenzo Librandi, Jul 06 2017

A036290 := proc(n) n*3^n ; end proc: # R. J. Mathar, Jun 18 2011

nn=20;a=1/(1-3x);CoefficientList[Series[x D[ a,x] ,{x,0,nn}],x]  (* Geoffrey Critzer, Nov 18 2012 *)
Table[n 3^n, {n, 0, 30}] (* Vincenzo Librandi, Jul 06 2017 *)

a(n)=3^n*n \\ Charles R Greathouse IV, Jun 18 2011

From Paul Barry, Feb 06 2004: (Start)
A trinomial transform. Differentiate (1+x+x^2)^n and set x=1.
a(n) = Sum_{i=0..n} Sum_{j=0..n} (2*n-2*i-j)*n!/(i!*j!*(n-i-j)!). (End)
From Paul Barry, Feb 15 2005: (Start)
a(n) = Sum_{k=0..2*n} T(n, k)*k, where T(n, k) is given by A027907.
a(n) = Sum_{k=0..n} Sum_{j=0..n} C(n, j)*C(j, k)*(j+k). (End)
From R. J. Mathar, Jun 19 2011: (Start)
G.f.: 3*x/(3*x-1)^2.
a(n) = 3*A027471(n+1). (End)
Sum_{n>=1} 1/a(n) = log(3/2) = 0.405465108... = A016578. - Franz Vrabec, Jan 07 2012
Sum_{n>=1} (-1)^(n+1)/a(n) = log(4/3) = A083679. - Amiram Eldar, Jul 20 2020
a(n) = 6*a(n-1) - 9*a(n-2). - Wesley Ivan Hurt, Apr 26 2021
From Elmo R. Oliveira, Sep 09 2024: (Start)
E.g.f.: 3*x*exp(3*x).
a(n) = n*A000244(n). (End)

A018215 a(n) = n*4^n.

Original entry on oeis.org

0, 4, 32, 192, 1024, 5120, 24576, 114688, 524288, 2359296, 10485760, 46137344, 201326592, 872415232, 3758096384, 16106127360, 68719476736, 292057776128, 1236950581248, 5222680231936, 21990232555520, 92358976733184, 387028092977152, 1618481116086272

Offset: 0

N. J. A. Sloane, Peter Winkler (pw(AT)bell-labs.com)

Bisection of A001787. That is, a(n) = A001787(2*n). - Graeme McRae, Jul 12 2006

All numbers of the form n*4^n+(4^n-1)/3 have the property that they are sums of two squares and also their indices are the sum of two squares. This follows from the identity n*4^n+(4^n-1)/3 = 4*(4*(..(4*(4*n+1)+1)..)+1)+1. - Artur Jasinski, Nov 12 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (8,-16).

Cf. A000302 (4^n), A001787, A002450, A083679.

Row n=4 of A258997.

[n*4^n: n in [0..25]]; // Vincenzo Librandi, Jun 01 2011

Table[n 4^n,{n,0,20}] (* or *) LinearRecurrence[{8,-16},{0,4},30] (* Harvey P. Dale, Apr 22 2018 *)

a(n) = n<<(2*n) \\ David A. Corneth, Apr 22 2018

G.f.: 4*x/(1-4*x)^2.
E.g.f.: 4*x*exp(4*x).
From Amiram Eldar, Jul 20 2020: (Start)
Sum_{n>=1} 1/a(n) = log(4/3) = A083679.
Sum_{n>=1} (-1)^(n+1)/a(n) = log(5/4). (End)

A016578 Decimal expansion of log(3/2).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

			0.4054651081081643819780131154643491365719904234624941976140143...

L. B. W. Jolley, Summation of Series, Dover (1961), eq (102), page 20.

Harry J. Smith, Table of n, a(n) for n = 0..20000
Index entries for transcendental numbers

Cf. A016529, A002391, A002162, A083679.

RealDigits[Log[3/2],10,111][[1]] (* Robert G. Wilson v, Aug 08 2011 *)

default(realprecision, 20080); x=10*log(3/2); for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b016578.txt", n, " ", d)); \\ Harry J. Smith, May 17 2009

Equals Sum {k>=1} 1/(k*3^k). - Robert G. Wilson v, Aug 08 2011
Equals 1/2 - 1/(2*2^2) + 1/(3*2^3) - 1/(4*2^4) + ... [Jolley].
Equals A002391-A002162. - Michel Marcus, Sep 17 2016
From Amiram Eldar, Aug 07 2020: (Start)
Equals 2 * arctanh(1/5).
Equals Integral_{x=0..oo} 1/(2*exp(x) + 1) dx. (End)
log(3/2) = 2*Sum_{n >= 1} 1/(n*P(n, 5)*P(n-1, 5)), where P(n, x) denotes the n-th Legendre polynomial. The first 10 terms of the series gives the approximation log(3/2) = 0.40546510810816438197(04...), correct to 20 decimal places. - Peter Bala, Mar 16 2024
Equals Sum_{n >= 1} (-1)^(n+1) * 5/(n*binomial(2*n, n)*6^n). The n-th term of the series is O(5*sqrt(Pi/n)*1/24^n). - Peter Bala, Mar 04 2025
Equals Integral_{x=0..1} (sqrt(x) - 1)/log(x) dx. - Kritsada Moomuang, Jun 14 2025

A248565 Least k such that log(4/3) - sum{1/(h*4^h), h = 1..k} < 1/8^n.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 16, 17, 19, 20, 21, 23, 24, 26, 27, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 43, 45, 46, 48, 49, 51, 52, 54, 55, 57, 58, 60, 61, 63, 64, 66, 67, 69, 70, 72, 73, 75, 76, 78, 79, 81, 82, 84, 85, 86, 88, 89, 91, 92, 94, 95

Offset: 1

Clark Kimberling, Oct 09 2014

This sequence provides insight into the manner of convergence of sum{1/(h*4^h), h = 1..k} to log(4/3). Since a(n+1) - a(n) is in {1,2} for n >= 1, the sequences A248566 and A248567 partition the positive integers.

			Let s(n) = log(4/3) - sum{1/(h*4^h), h = 1..n}.  Approximations follow:
n ... s(n) ........ 1/8^n
1 ... 0.037682 ... 0.125
2 ... 0.006432 ... 0.015625
3 ... 0.001223 ... 0.001953
4 ... 0.000247 ... 0.000244
5 ... 0.000051 ... 0.000030
a(4) = 5 because s(5) < 1/8^4 < s(4).

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 15.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A083679 (log(4/3)), A248566, A248567, A248559, A248565.

z = 2500; p[k_] := p[k] = Sum[1/(h*4^h), {h, 1, k}];
N[Table[p[k], {k, 1, z/5}], 12];
N[Table[Log[4/3] - p[n], {n, 1, z/5}]];
f[n_] := f[n] = Select[Range[z], Log[4/3] - p[#] < 1/8^n &, 1];
u = Flatten[Table[f[n], {n, 1, z}]] ; (* A248565 *)
Flatten[Position[Differences[u], 1]]; (* A248566 *)
Flatten[Position[Differences[u], 2]]; (* A248567 *)

A069864 Decimal expansion of 2/log(4/3).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jun 24 2002

Constant arising in analysis of Kolakoski sequence A000002.

			6.95211899356441...

Jeffrey C. Lagarias and K. Soundararajan, Benford's law for the 3x+1 function, arXiv:math/0509175 [math.NT], 2005-2006.
Index entries for transcendental numbers.

Cf. A000002, A083679 (log(4/3)).

SetDefaultRealField(RealField(100));  2 / Log(4 / 3); // Vincenzo Librandi, May 16 2019

RealDigits[N[2 / Log[4 / 3], 150]][[1]] (* Vincenzo Librandi, May 16 2019 *)

2/log(4/3) \\ Charles R Greathouse IV, May 15 2019

a(99) corrected by Georg Fischer, Apr 03 2020

A369522 Decimal expansion of log(4/3) / log(3/2).

Original entry on oeis.org

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

Offset: 0

Ruud H.G. van Tol, Jan 25 2024

			0.70951129135145477697619...

Index entries for transcendental numbers

Cf. A016578, A083679, A368131.

Join[{0},First[RealDigits[Log[4/3]/Log[3/2],10,86]]] (* James C. McMahon, Jan 30 2024 *)

PARI
```
(log(4)-log(3)) / (log(3)-log(2))
```

Equals A083679 / A016578.

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A036290 a(n) = n*3^n.

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A018215 a(n) = n*4^n.

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A016578 Decimal expansion of log(3/2).

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A248565 Least k such that log(4/3) - sum{1/(h*4^h), h = 1..k} < 1/8^n.

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A069864 Decimal expansion of 2/log(4/3).

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A369522 Decimal expansion of log(4/3) / log(3/2).

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