A024023 -id:A024023 - cp's OEIS frontend

A027871 a(n) = Product_{i=1..n} (3^i - 1).

1, 2, 16, 416, 33280, 8053760, 5863137280, 12816818094080, 84078326697164800, 1654829626053597593600, 97714379759212830706892800, 17309711516825516108403231948800

Offset: 0

N. J. A. Sloane

nonn

2*(10)^m|a(n) where 4*m <= n <= 4*m+3 for m >= 1. - G. C. Greubel, Nov 20 2015

Given probability p = 1/3^n that an outcome will occur at the n-th stage of an infinite process, then starting at n=1, 1-a(n)/A047656(n+1) is the probability that the outcome has occurred at or before the n-th iteration. The limiting ratio is 1-A100220 ~ 0.4398739. - Bob Selcoe, Mar 01 2016

G. C. Greubel, Table of n, a(n) for n = 0..50

Cf. A005329 (q=2), A027637 (q=4), A027872 (q=5), A027873 (q=6), A027875 (q=7), A027876 (q=8), A027877 (q=9), A027878 (q=10), A027879 (q=11), A027880 (q=12).

Cf. A047656, A100220, A132324.

[1] cat [&*[ 3^k-1: k in [1..n] ]: n in [1..11]]; // Vincenzo Librandi, Dec 24 2015

A027871 := proc(n)
    mul( 3^i-1,i=1..n) ;
end proc:
seq(A027871(n),n=0..8) ; # R. J. Mathar, Jul 13 2017

Table[Product[(3^k-1),{k,1,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 17 2015 *)
Abs@QPochhammer[3, 3, Range[0, 10]] (* Vladimir Reshetnikov, Nov 20 2015 *)

a(n) = prod(i=1, n, 3^i-1); \\ Michel Marcus, Nov 21 2015

a(n) ~ c * 3^(n*(n+1)/2), where c = A100220 = Product_{k>=1} (1-1/3^k) = 0.560126077927948944969792243314140014379736333798... . - Vaclav Kotesovec, Nov 21 2015
a(n) = 3^(binomial(n+1,2))*(1/3;1/3){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Dec 24 2015
a(n) = Product_{i=1..n} A024023(i). - Michel Marcus, Dec 27 2015
G.f.: Sum_{n>=0} 3^(n*(n+1)/2)*x^n / Product_{k=0..n} (1 + 3^k*x). - Ilya Gutkovskiy, May 22 2017
From Amiram Eldar, Feb 19 2022: (Start)
Sum_{n>=0} 1/a(n) = A132324.
Sum_{n>=0} (-1)^n/a(n) = A100220. (End)

A059379 Array of values of Jordan function J_k(n) read by antidiagonals (version 1).

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 2, 8, 7, 1, 4, 12, 26, 15, 1, 2, 24, 56, 80, 31, 1, 6, 24, 124, 240, 242, 63, 1, 4, 48, 182, 624, 992, 728, 127, 1, 6, 48, 342, 1200, 3124, 4032, 2186, 255, 1, 4, 72, 448, 2400, 7502, 15624, 16256, 6560, 511, 1, 10, 72, 702, 3840

Offset: 1

N. J. A. Sloane, Jan 28 2001

			Array begins:
  1,  1,  2,   2,   4,    2,    6,    4,   6,  4, 10, 4, ...
  1,  3,  8,  12,  24,   24,   48,   48,  72, 72, ...
  1,  7, 26,  56, 124,  182,  342,  448, 702, ...
  1, 15, 80, 240, 624, 1200, 2400, 3840, ...

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
R. Sivaramakrishnan, "The many facets of Euler's totient. II. Generalizations and analogues", Nieuw Arch. Wisk. (4) 8 (1990), no. 2, 169-187.

Enrique Pérez Herrero, Table of n, a(n) for n = 1..10000

See A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A059376 (J_3), A059377 (J_4), A059378 (J_5). Columns give A000225, A024023, A020522, A024049, A059387, etc.

Main diagonal gives A067858.

J := proc(n,k) local i,p,t1,t2; t1 := n^k; for p from 1 to n do if isprime(p) and n mod p = 0 then t1 := t1*(1-p^(-k)); fi; od; t1; end;
#alternative
A059379 := proc(n,k)
    add(d^k*numtheory[mobius](n/d),d=numtheory[divisors](n)) ;
end proc:
seq(seq(A059379(d-k,k),k=1..d-1),d=2..12) ; # R. J. Mathar, Nov 23 2018

JordanTotient[n_,k_:1]:=DivisorSum[n,#^k*MoebiusMu[n/#]&]/;(n>0)&&IntegerQ[n];
A004736[n_]:=Binomial[Floor[3/2+Sqrt[2*n]],2]-n+1;
A002260[n_]:=n-Binomial[Floor[1/2+Sqrt[2*n]],2];
A059379[n_]:=JordanTotient[A004736[n],A002260[n]]; (* Enrique Pérez Herrero, Dec 19 2010 *)

jordantot(n,k)=sumdiv(n,d,d^k*moebius(n/d));
A002260(n)=n-binomial(floor(1/2+sqrt(2*n)),2);
A004736(n)=binomial(floor(3/2+sqrt(2*n)),2)-n+1;
A059379(n)=jordantot(A004736(n),A002260(n)); \\ Enrique Pérez Herrero, Jan 08 2011

from functools import cache
def MoebiusTrans(a, i):
    @cache
    def mb(n, d = 1):
          return d % n and -mb(d, n % d < 1) + mb(n, d + 1) or 1 // n
    def mob(m, n): return mb(m // n) if m % n == 0 else 0
    return sum(mob(i, d) * a(d) for d in range(1, i + 1))
def Jrow(n, size):
    return [MoebiusTrans(lambda m: m ** n, k) for k in range(1, size)]
for n in range(1, 8): print(Jrow(n, 13))
# Alternatively:
from sympy import primefactors as prime_divisors
from fractions import Fraction as QQ
from math import prod as product
def J(n: int, k: int) -> int:
    t = QQ(pow(k, n), 1)
    s = product(1 - QQ(1, pow(p, n)) for p in prime_divisors(k))
    return (t * s).numerator  # the denominator is always 1
for n in range(1, 8): print([J(n, k) for k in range(1, 13)])
# Peter Luschny, Dec 16 2023

J_k(n) = Sum_{d|n} d^k*mu(n/d). - Benoit Cloitre and Michael Orrison (orrison(AT)math.hmc.edu), Jun 07 2002
From Amiram Eldar, Jun 07 2025: (Start)
For a given k, J_k(n) is multiplicative with J_k(p^e) = p^(k*e) - p^(k*e-k).
For a given k, Dirichlet g.f. of J_k(n): zeta(s-k)/zeta(s).
Sum_{i=1..n} J_k(i) ~ n^(k+1) / ((k+1)*zeta(k+1)).
Sum_{n>=1} 1/J_k(n) = Product_{p prime} (1 + p^k/(p^k-1)^2) for k >= 2. (End)

A057958 Number of prime factors of 3^n - 1 (counted with multiplicity).

Original entry on oeis.org

1, 3, 2, 5, 3, 5, 2, 7, 3, 6, 3, 8, 2, 5, 5, 10, 3, 8, 3, 10, 4, 7, 3, 11, 5, 5, 6, 9, 4, 11, 4, 12, 5, 8, 6, 12, 3, 7, 7, 13, 4, 11, 3, 11, 9, 6, 5, 17, 7, 10, 6, 9, 4, 13, 8, 13, 7, 9, 3, 17, 3, 8, 6, 14, 7, 12, 4, 12, 6, 11, 2, 16, 5, 8, 10, 11, 7, 15, 4, 18, 9, 8, 5, 18, 7, 6, 8, 16, 4, 19, 5

Offset: 1

Patrick De Geest, Nov 15 2000

nonn

Max Alekseyev, Table of n, a(n) for n = 1..690 (first 660 terms from Amiram Eldar)
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n-1): A057951 (b=10), A057952 (b=9), A057953 (b=8), A057954 (b=7), A057955 (b=6), A057956 (b=5), A057957 (b=4), this sequence (b=3), A046051 (b=2).

Cf. A001222, A024023, A085028, A002591 A074477, A057952, A133801, A274909.

a(n)=bigomega(3^n-1) \\ Charles R Greathouse IV, Sep 14 2015

Mobius transform of A085028. - T. D. Noe, Jun 19 2003

a(n) = A001222(A024023(n)). - Amiram Eldar, Feb 01 2020

Offset corrected by Amiram Eldar, Feb 01 2020

A059380 Array of values of Jordan function J_k(n) read by antidiagonals (version 2).

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 7, 8, 2, 1, 15, 26, 12, 4, 1, 31, 80, 56, 24, 2, 1, 63, 242, 240, 124, 24, 6, 1, 127, 728, 992, 624, 182, 48, 4, 1, 255, 2186, 4032, 3124, 1200, 342, 48, 6, 1, 511, 6560, 16256, 15624, 7502, 2400, 448, 72, 4, 1, 1023, 19682

Offset: 1

N. J. A. Sloane, Jan 28 2001

			Array begins:
1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, ...
1, 3, 8, 12, 24, 24, 48, 48, 72, 72, ...
1, 7, 26, 56, 124, 182, 342, 448, 702, ...
1, 15, 80, 240, 624, 1200, 2400, 3840, ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
R. Sivaramakrishnan, The many facets of Euler's totient. II. Generalizations and analogues, Nieuw Arch. Wisk. (4) 8 (1990), no. 2, 169-187

Enrique Pérez Herrero, Table of n, a(n) for n = 1..10000

See A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A059376 (J_3), A059377 (J_4), A059378 (J_5). Columns give A000225, A024023, A020522, A024049, A059387, etc.

Main diagonal gives A067858.

J := proc(n,k) local i,p,t1,t2; t1 := n^k; for p from 1 to n do if isprime(p) and n mod p = 0 then t1 := t1*(1-p^(-k)); fi; od; t1; end;

JordanTotient[n_,k_:1]:=DivisorSum[n,#^k*MoebiusMu[n/#]&]/;(n>0)&&IntegerQ[n];
A004736[n_]:=Binomial[Floor[3/2+Sqrt[2*n]],2]-n+1;
A002260[n_]:=n-Binomial[Floor[1/2+Sqrt[2*n]],2];
A059380[n_]:=JordanTotient[A002260[n],A004736[n]]; (* Enrique Pérez Herrero, Dec 19 2010 *)

jordantot(n,k)=sumdiv(n,d,d^k*moebius(n/d));
A002260(n)=n-binomial(floor(1/2+sqrt(2*n)),2);
A004736(n)=binomial(floor(3/2+sqrt(2*n)),2)-n+1;
A059380(n)=jordantot(A002260(n),A004736(n)); \\ Enrique Pérez Herrero, Jan 08 2011

A067771 Number of vertices in Sierpiński triangle of order n.

Original entry on oeis.org

3, 6, 15, 42, 123, 366, 1095, 3282, 9843, 29526, 88575, 265722, 797163, 2391486, 7174455, 21523362, 64570083, 193710246, 581130735, 1743392202, 5230176603, 15690529806, 47071589415, 141214768242, 423644304723, 1270932914166

Offset: 0

Martin Wessendorf (martinw(AT)mail.ahc.umn.edu), Feb 09 2002

An order 0 Sierpiński triangle is a triangle. Order n+1 is formed from three copies of order n by identifying pairs of corner vertices of each pair of triangles. - Allan Bickle, Aug 03 2024
This sequence represents another link from the product factor space Q X Q / {(1,1), (-1, -1)} to Sierpiński's triangle. The first "link" found was to sequence A048473. - Creighton Dement, Aug 05 2004
a(n) equals the number of orbits of the finite group PSU(3,3^n) on subsets of size 3 of the 3^(3n)+1 isotropic points of a unitary 3 space. - Paul M. Bradley, Jan 31 2017
For n >= 1, number of edges in a planar Apollonian graph at iteration n. - Andrew D. Walker, Jul 08 2017
Also the total domination number of the (n+3)-Dorogovtsev-Goltsev-Mendes graph, using the convention DGM(0) = P_2. - Eric W. Weisstein, Jan 14 2024

			Order 0 is a triangle, so a(0) = 3.
Order 1 has three corners (degree 2) and three other vertices, so a(1) = 6.
3 example graphs:                        o
                                        / \
                                       o---o
                                      / \ / \
                        o            o---o---o
                       / \          / \     / \
             o        o---o        o---o   o---o
            / \      / \ / \      / \ / \ / \ / \
           o---o    o---o---o    o---o---o---o---o
Graph:      S_1        S_2              S_3
Vertices:    3          6                15
Edges:       3          9                27

Peter Wessendorf and Kristina Downing, personal communication.

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Allan Bickle, Properties of Sierpinski Triangle Graphs, Springer PROMS 448 (2021) 295-303.
Paul Bradley and Peter Rowley, Orbits on k-subsets of 2-transitive Simple Lie-type Groups, 2014.
A. Hinz, S. Klavzar, and S. Zemljic, A survey and classification of Sierpinski-type graphs, Discrete Applied Mathematics 217 3 (2017), 565-600.
András Kaszanyitzky, Triangular fractal approximating graphs and their covering paths and cycles, arXiv:1710.09475 [math.CO], 2017. See Table 2.
C. Lanius, Fractals.
Eric Weisstein's World of Mathematics, Dorogovtsev-Goltsev-Mendes Graph.
Eric Weisstein's World of Mathematics, Sierpiński Gasket Graph.
Eric Weisstein's World of Mathematics, Total Domination Number.
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A048473.
Cf. A003462, A007051, A034472, A024023.
Cf. A007283, A029858, A067771, A193277, A233774, A233775, A246959 (Sierpiński triangle graphs).

[(3/2)*(1+3^n): n in [0..30]]; // Vincenzo Librandi, Jun 20 2011

LinearRecurrence[{4, -3}, {3, 6}, 26] (* or *)
CoefficientList[Series[3 (1 - 2 x)/((1 - x) (1 - 3 x)), {x, 0, 25}], x] (* Michael De Vlieger, Feb 02 2017 *)
Table[3/2 (3^n + 1), {n, 0, 20}] (* Eric W. Weisstein, Jan 14 2024 *)
3/2 (3^Range[0, 20] + 1) (* Eric W. Weisstein, Jan 14 2024 *)

a(n) = (3/2)*(3^n + 1).
a(n) = 3 + 3^1 + 3^2 + 3^3 + 3^4 + ... + 3^n = 3 + Sum_{k=1..n} 3^n.
a(n) = 3*A007051(n).
a(0) = 3, a(n) = a(n-1) + 3^n. a(n) = (3/2)*(1+3^n). - Zak Seidov, Mar 19 2007
a(n) = 4*a(n-1) - 3*a(n-2).
G.f.: 3*(1-2*x)/((1-x)*(1-3*x)). - Colin Barker, Jan 10 2012
a(n) = A233774(2^n). - Omar E. Pol, Dec 16 2013
a(n) = 3*a(n-1) - 3. - Zak Seidov, Oct 26 2014
E.g.f.: 3*(exp(x) + exp(3*x))/2. - Stefano Spezia, Feb 09 2021
a(n) = A029858(n+1) + 3. - Allan Bickle, Aug 03 2024

More terms from Benoit Cloitre, Feb 22 2002

A100774 a(n) = 2*(3^n - 1).

Original entry on oeis.org

0, 4, 16, 52, 160, 484, 1456, 4372, 13120, 39364, 118096, 354292, 1062880, 3188644, 9565936, 28697812, 86093440, 258280324, 774840976, 2324522932, 6973568800, 20920706404, 62762119216, 188286357652, 564859072960, 1694577218884

Offset: 0

Pawel P. Mazur (Pawel.Mazur(AT)pwr.wroc.pl), Apr 06 2005

a(n) is the number of steps which are made when generating all n-step nonreversing random walks that begin in a fixed point P on a two-dimensional square lattice. To make one step means to move along one edge on the lattice.
These are also the first local maxima reached in the Collatz trajectories of 2^n - 1. - David Rabahy, Oct 30 2017
Also the graph diameter of the n-Sierpinski carpet graph. - Eric W. Weisstein, Mar 13 2018
a(n) is the number of edge covers of F_{n,2}, which has adjacent vertices u and w, and n vertices each adjacent to both u and w. An edge cover is a subset of the edges where each vertex is adjacent to at least one vertex. To cover each of the n vertices v_i, we need to have at least the edge uv_i or wv_i or both, giving us three choices for each. We can then add the edge uw or not, which is 2*3^n choices. But we need to remove the case where all uv_i's were chosen and uw not chosen, and all ww_i's were chosen and uw not chosen. - Feryal Alayont, Jun 17 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..196
Eric Weisstein's World of Mathematics, Graph Diameter
Eric Weisstein's World of Mathematics, Sierpinski Carpet Graph
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A003462, A007051, A024023, A029858, A034472, A048473, A058481, A067771, A079004, A115099, A134931.

[2*(3^n - 1): n in [0..25] ]; // Vincenzo Librandi, Apr 30 2011

Table[2 (3^n - 1), {n, 0, 24}] (* Alonso del Arte, Nov 08 2012 *)
2 (3^Range[0, 20] - 1) (* Eric W. Weisstein, Mar 13 2018 *)
LinearRecurrence[{4, -3}, {4, 16}, {0, 20}] (* Eric W. Weisstein, Mar 13 2018 *)
CoefficientList[Series[4 x/(1 - 4 x + 3 x^2), {x, 0, 20}], x] (* Eric W. Weisstein, Mar 13 2018 *)

A100774(n):=2*(3^n - 1)$
makelist(A100774(n),n,0,30); /* Martin Ettl, Nov 09 2012 */

a(n)=2*3^n-2 \\ Charles R Greathouse IV, Nov 09 2012

a(n) = 2*(3^n - 1).
a(n) = 4*Sum_{i=0..n-1} 3^i.
a(n) = 4*A003462(n).
a(n) = A048473(n) - 1. - Paul Curtz, Jan 19 2009
G.f.: 4*x/((1-x)*(1-3*x)). - Eric W. Weisstein, Mar 13 2018
a(n) = 4*a(n-1) - 3*a(n-2). - Eric W. Weisstein, Mar 13 2018
From Elmo R. Oliveira, Dec 06 2023: (Start)
a(n) = 2*A024023(n).
a(n) = 3*a(n-1) + 4 for n>0.
E.g.f.: 2*(exp(3*x) - exp(x)). (End)

A133801 Number of distinct prime divisors of 3^n - 1.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 2, 3, 3, 3, 3, 5, 2, 3, 4, 5, 3, 6, 3, 5, 4, 5, 3, 7, 4, 3, 6, 6, 4, 8, 4, 6, 5, 6, 5, 9, 3, 5, 6, 7, 4, 8, 3, 8, 8, 4, 5, 12, 7, 7, 6, 6, 4, 11, 6, 9, 7, 7, 3, 12, 3, 6, 6, 7, 6, 10, 4, 9, 6, 8, 2, 12, 5, 6, 9, 8, 7, 12, 4, 11, 9, 6, 5, 14, 6, 4, 8, 12, 4, 16, 5, 7, 7, 8, 6, 15, 4, 10, 8

Offset: 1

Ryan Propper, Jan 06 2008

nonn

			a(4) = omega(3^4 - 1) = omega(80) = omega(2^4 * 5) = 2.

Table of n, a(n) for n = 1..690
Eric Weisstein's World of Mathematics, Prime Factor.

Cf. A001221, A024023, A113913.

Table[PrimeNu[3^n - 1], {n, 1, 50}] (* G. C. Greubel, May 21 2017 *)

for(n = 1, 100, print1(omega(3^n - 1), ", "))

a(n) = omega(3^n - 1) = A001221(3^n - 1).

Terms to a(660) in b-file from Amiram Eldar, Feb 03 2020

a(661)-a(690) in b-file from Max Alekseyev, May 22 2022

A134931 a(n) = (5*3^n-3)/2.

Original entry on oeis.org

1, 6, 21, 66, 201, 606, 1821, 5466, 16401, 49206, 147621, 442866, 1328601, 3985806, 11957421, 35872266, 107616801, 322850406, 968551221, 2905653666, 8716961001, 26150883006, 78452649021, 235357947066, 706073841201, 2118221523606

Offset: 0

Rolf Pleisch, Jan 29 2008

Numbers n where the recurrence s(0)=1, if s(n-1) >= n then s(n) = s(n-1) - n else s(n) = s(n-1) + n produces s(n)=0. - Hugo Pfoertner, Jan 05 2012
A046901(a(n)) = 1. - Reinhard Zumkeller, Jan 31 2013
Binomial transform of A146523: (1, 5, 10, 20, 40, ...) and double binomial transform of A010685: (1, 4, 1, 4, 1, 4, ...). - Gary W. Adamson, Aug 25 2016
Also the number of maximal cliques in the (n+1)-Hanoi graph. - Eric W. Weisstein, Dec 01 2017
a(n) is the least k such that f(a(n-1)+1) + ... + f(k) > f(a(n-2)+1) + ... + f(a(n-1)) for n > 1, where f(n) = 1/(n+1). Because Sum_{k=1..5*3^(n-1)} 1/(a(n)+3*k-1) + 1/(a(n)+3*k) + 1/(a(n)+3*k+1) - 1/((a(n)+1+5*3^n)*5*3^(n-1)) < Sum_{k=1..5*3^(n-1)} 1/(a(n-1)+k+1) < Sum_{k=1..5*3^(n-1)} 1/(a(n)+3*k-1) + 1/(a(n)+3*k) + 1/(a(n)+3*k+1), we have 1 < 1/3 + 1/4 + ... + 1/7 < 1/8 + 1/9 + ... + 1/22 < ... . - Jinyuan Wang, Jun 15 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Hanoi Graph
Eric Weisstein's World of Mathematics, Maximal Clique
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A003462, A007051, A034472, A024023, A067771, A029858.

Cf. A005030, A010685, A046901, A060816, A116952, A146523, A225918.

[(5*3^n-3)/2: n in [0..30]]; // Vincenzo Librandi, Jun 05 2011

seq((5*3^n-3)/2, n= 0..25); # Gary Detlefs, Jun 22 2010

a=1; lst={a}; Do[a=a*3+3; AppendTo[lst,a], {n,0,100}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 25 2008 *)
Table[(5 3^n - 9)/6, {n, 20}] (* Eric W. Weisstein, Dec 01 2017 *)
(5 3^Range[20] - 9)/6 (* Eric W. Weisstein, Dec 01 2017 *)
LinearRecurrence[{4, -3}, {1, 6}, 20] (* Eric W. Weisstein, Dec 01 2017 *)
CoefficientList[Series[(1 + 2 x)/(1 - 4 x + 3 x^2), {x, 0, 20}], x] (* Eric W. Weisstein, Dec 01 2017 *)

a(n) = (5*3^n-3)/2; /* Joerg Arndt, Apr 14 2013 */

a(n) = 3*(a(n-1) + 1), with a(0)=1.
From R. J. Mathar, Jan 31 2008: (Start)
O.g.f.: (5/2)/(1-3*x) - (3/2)/(1-x).
a(n) = (A005030(n) - 3)/2. (End)
a(n) = A060816(n+1) - 1. - Philippe Deléham, Apr 14 2013
E.g.f.: exp(x)*(5*exp(2*x) - 3)/2. - Stefano Spezia, Aug 28 2023

More terms from Vladimir Joseph Stephan Orlovsky, Dec 25 2008

A168607 a(n) = 3^n + 2.

Original entry on oeis.org

3, 5, 11, 29, 83, 245, 731, 2189, 6563, 19685, 59051, 177149, 531443, 1594325, 4782971, 14348909, 43046723, 129140165, 387420491, 1162261469, 3486784403, 10460353205, 31381059611, 94143178829, 282429536483, 847288609445

Offset: 0

Vincenzo Librandi, Dec 01 2009

Second bisection is A134752.
It appears that if s(n) is a first order rational sequence of the form s(1)=5, s(n)= (2*s(n-1)+1)/(s(n-1)+2),n>1, then s(n)= a(n)/(a(n)-4), n>1. - Gary Detlefs, Nov 16 2010
Mahler exhibits this sequence with n>=1 as a proof that there exists an infinite number of x coprime to 3, such that x belongs to A125293 and x^2 belongs to A005836. - Michel Marcus, Nov 12 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Gennady Eremin, Arithmetization of well-formed parenthesis strings. Motzkin Numbers of the Second Kind, arXiv:2012.12675 [math.CO], 2020.
Kurt Mahler, The representation of squares to the base 3, Acta Arith. Vol. 53, Issue 1 (1989), p. 99-106.
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A008776 (2*3^n), A005051 (8*3^n), A034472 (3^n+1), A000244 (powers of 3), A024023 (3^n-1), A168609 (3^n+4), A168610 (3^n+5), A134752 (3^(2*n-1)+2).

Magma
```
[3^n+2: n in [0..30]];
```

A168607:=n->3^n + 2; seq(A168607(n), n=0..30); # Wesley Ivan Hurt, Mar 21 2014

CoefficientList[Series[(3 - 7 x)/((1-x) (1-3 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Feb 06 2013 *)
NestList[3 # - 4 & , 3, 25] (* Bruno Berselli, Feb 06 2013 *)

a(n)=3^n+2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 3*a(n-1) - 4, a(0) = 3.
a(n+1) - a(n) = A008776(n).
a(n+2) - a(n) = A005051(n).
a(n) = A034472(n)+1 = A000244(n)+2 = A024023(n)+3 = A168609(n)-2 = A168610(n)-3.
G.f.: (3 - 7*x)/((1 - x)*(1 - 3*x)).
a(n) = 4*a(n-1) - 3*a(n-2), a(0) = 3, a(1) = 5. - Vincenzo Librandi, Feb 06 2013
E.g.f.: exp(3*x) + 2*exp(x). - Elmo R. Oliveira, Nov 09 2023

Edited by Klaus Brockhaus, Apr 13 2010

Further edited by N. J. A. Sloane, Aug 10 2010

A214369 Decimal expansion of Sum_{n>=1} 1/(3^n-1).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jul 14 2012

			Equals 0.6821535026052380667...

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A024023, A065442, A073668, A248721, A248722, A248723, A248724, A248725, A248726.

evalf(sum(1/(3^k-1), k=1..infinity), 120); # Vaclav Kotesovec, Oct 18 2014
# second program with faster converging series
evalf( add( (1/3)^(n^2)*(1 + 2/(3^n - 1)), n = 1..14 ), 105); # Peter Bala, Jan 30 2022

RealDigits[ NSum[1/(3^n - 1), {n, 1, Infinity}, WorkingPrecision -> 110, NSumTerms -> 100], 10, 105] // First (* or *) 1 - (Log[2] + QPolyGamma[0, 1, 1/3])/Log[3] // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Jun 05 2013 *)
x = 1/3; RealDigits[ Sum[ DivisorSigma[0, k] x^k, {k, 1000}], 10, 105][[1]] (* Robert G. Wilson v, Oct 12 2014 after an observation and the formula of Amarnath Murthy, see A073668 *)

suminf(n=1, 1/(3^n-1)) \\ Michel Marcus, Mar 11 2017

Equals Sum_{n>=1} 1/A024023(n).

Equals Sum_{k>=1} d(k)/3^k, where d(k) is the number of divisors of k (A000005). - Amiram Eldar, May 17 2020

More terms from Jean-François Alcover, Feb 12 2013

cp's OEIS Frontend

A027871 a(n) = Product_{i=1..n} (3^i - 1).

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A059379 Array of values of Jordan function J_k(n) read by antidiagonals (version 1).

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A057958 Number of prime factors of 3^n - 1 (counted with multiplicity).

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A059380 Array of values of Jordan function J_k(n) read by antidiagonals (version 2).

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A067771 Number of vertices in Sierpiński triangle of order n.

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A100774 a(n) = 2*(3^n - 1).

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A133801 Number of distinct prime divisors of 3^n - 1.

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