A097934 -id:A097934 - cp's OEIS frontend

A001047 a(n) = 3^n - 2^n.

0, 1, 5, 19, 65, 211, 665, 2059, 6305, 19171, 58025, 175099, 527345, 1586131, 4766585, 14316139, 42981185, 129009091, 387158345, 1161737179, 3485735825, 10458256051, 31376865305, 94134790219, 282412759265, 847255055011, 2541798719465, 7625463267259, 22876524019505

Offset: 0

N. J. A. Sloane, R. K. Guy

a(n+1) is the sum of the elements in the n-th row of triangle pertaining to A036561. - Amarnath Murthy, Jan 02 2002
Number of 2 X n binary arrays with a path of adjacent 1's and no path of adjacent 0's from top row to bottom row. - R. H. Hardin, Mar 21 2002
With offset 1, partial sums of A027649. - Paul Barry, Jun 24 2003
Number of distinct lines through the origin in the n-dimensional lattice of side length 2. A049691 has the values for the 2-dimensional lattice of side length n. - Joshua Zucker, Nov 19 2003
a(n+1)/(n+1)=(3*3^n-2*2^n)/(n+1) is the second binomial transform of the harmonic sequence 1/(n+1). - Paul Barry, Apr 19 2005
a(n+1) is the sum of n-th row of A036561. - Reinhard Zumkeller, May 14 2006
The sequence gives the sum of the lengths of the segments in Cantor's dust generating sequence up to the i-th step. Measurement unit = length of the segment of i-th step. - Giorgio Balzarotti, Nov 18 2006
Let T be a binary relation on the power set P(A) of a set A having n = |A| elements such that for every element x, y of P(A), xTy if x is a proper subset of y. Then a(n) = |T|. - Ross La Haye, Dec 22 2006
From Alexander Adamchuk, Jan 04 2007: (Start)
a(n) is prime for n in A057468.
p divides a(p) - 1 for prime p.
Quotients (3^p - 2^p - 1)/p, where p = prime(n), are listed in A127071.
Numbers k such that k divides 3^k - 2^k - 1 are listed in A127072.
Pseudoprimes in A127072(n) include all powers of primes {2,3,7} and some composite numbers that are listed in A127073, which includes all Carmichael numbers A002997.
Numbers n such that n^2 divides 3^n - 2^n - 1 are listed in A127074.
5 divides a(2n).
5^2 divides a(2*5n).
5^3 divides a(2*5^2n).
5^4 divides a(2*5^3n).
7^2 divides a(6*7n).
13 divides a(4n).
13^2 divides a(4*13n).
19 divides a(3n).
19^2 divides a(3*19n).
23^2 divides a(11n).
23^3 divides a(11*23n).
23^4 divides a(11*23^2n).
29 divides a(7n).
p divides a((p-1)n) for prime p>3.
p divides a((p-1)/2) for prime p in A097934. Also primes p such that 6 is a square mod p, except {2,3}, A038876(n).
p^(k+1) divides a(p^k*(p-1)/2*n) for prime p in A097934.
p^(k+1) divides a(p^k*(p-1)*n) for prime p>3.
Note the exception that for p = 23, p^(k+2) divides a(p^k*(p-1)/2*n).
There are no more such exceptions for primes p up to 600000. (End)
a(n) divides a(q*(n+1)-1), for all q integer. Leonardo Sarasua, Apr 15 2024
Final digits of terms follow sequence 1,5,9,5. - Enoch Haga, Nov 26 2007
This is also the second column sequence of the Sheffer triangle A143494 (2-restricted Stirling2 numbers). See the e.g.f. given below. - Wolfdieter Lang, Oct 08 2011
Partial sums give A000392. - Jon Perry, Apr 05 2014
For n >= 1, this is also row 2 of A281890: when consecutive positive integers are written as a product of primes in nondecreasing order, "3" occurs in n-th position a(n) times out of every 6^n. - Peter Munn, May 17 2017
a(n) is the number of ternary sequences of length n which include the digit 2. For example, a(2)=5 since the sequences are 02,20,12,21,22. - Enrique Navarrete, Apr 05 2021
a(n-1) is the number of ways we can form disjoint unions of two nonempty subsets of [n] such that the union contains n. For example, for n = 3, a(2) = 5 since the  disjoint unions are {1}U{3}, {1}U{2,3}, {2}U{3}, {2}U{1,3}, and {1,2}U{3}. Cf. A000392 if we drop the requirement that the union contains n. - Enrique Navarrete, Aug 24 2021
Configures as a composite Koch Snowflake Fractal (see illustration in links) based on the five-fold division of the Cantor Square/Cantor Dust Fractal of (9^n-4^n)/5 see my illustration in (A016153). - John Elias, Oct 13 2021
Number of pairs (A,B) where B is a subset of {1,2,...,n} and A is a proper subset of B. - Jianing Song, Jun 18 2022
From Manfred Boergens, Mar 29 2023: (Start)
With regard to the comments by Ross La Haye and Jianing Song: Omitting "proper" gives A000244.
Number of pairs (A,B) where B is a nonempty subset of {1,2,...,n} and A is a nonempty subset of B. For nonempty proper subsets see a(n+1) in A028243. (End)
a(n) is the number of n-digit numbers whose smallest decimal digit is 7. - Stefano Spezia, Nov 15 2023
a(n-1) is the number of all possible player-reduced binary games observed by each player in an nx2 game assuming the individual strategies of k < n - 1 players are fixed and the remaining n - k - 1 player will play as one, either maintaining their status quo strategies or jointly adopting an alternative strategy. - Ambrosio Valencia-Romero, Apr 11 2024

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 86-87.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..200
A. Abdurrahman, CM Method and Expansion of Numbers, arXiv:1909.10889 [math.NT], 2019.
Nathan Bliss, Ben Fulan, Stephen Lovett and Jeff Sommars, Strong divisibility, cyclotomic polynomials and iterated polynomials, Am. Math. Monthly, Vol. 120, No. 6 (2013), pp. 519-536.
John Elias, Illustration: Sierpinski half-hexagons, Illustration: Nicomachus triangle 2^n & 3^n correlation, Koch Snowflake Fractal Configuration.
Joël Gay, Representation of Monoids and Lattice Structures in the Combinatorics of Weyl Groups, Doctoral Thesis, Discrete Mathematics [cs.DM], Université Paris-Saclay, 2018.
Samuele Giraudo, Combinatorial operads from monoids, Journal of Algebraic Combinatorics, Vol. 41, No. 2 (2015), pp. 493-538; arXiv preprint, arXiv preprint arXiv:1306.6938 [math.CO], 2013-2015.
Samuele Giraudo, Pluriassociative algebras I: The pluriassociative operad, Advances in Applied Mathematics, Vol. 77 (2016), pp. 1-42; arXiv preprint, arXiv:1603.01040 [math.CO], 2016.
Richard K. Guy, Letters to N. J. A. Sloane, June-August 1968
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 397.
B. D. Josephson and J. M. Boardman, Problems Drive 1961, Eureka, The Journal of the Archimedeans, Vol. 24 (1961), p. 20; entire volume.
Germain Kreweras, Inversion des polynômes de Bell bidimensionnels et application au dénombrement des relations binaires connexes, C. R. Acad. Sci. Paris Ser. A-B, Vol. 268 (1969), pp. A577-A579.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Richard Miles, Synchronization points and associated dynamical invariants, Trans. Amer. Math. Soc., Vol. 365, No. 10 (2013), pp. 5503-5524.
Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics (2022) Vol. 10, No. 7, 1161.
Jon Perry, Relation to Collatz problem.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Kalika Prasad, Munesh Kumari, Rabiranjan Mohanta, and Hrishikesh Mahato, The sequence of higher order Mersenne numbers and associated binomial transforms, arXiv:2307.08073 [math.NT], 2023.
D. C. Santos, E. A. Costa, and P. M. M. C. Catarino, On Gersenne Sequence: A Study of One Family in the Horadam-Type Sequence, Axioms 14, 203, (2025). See p. 4.
Ambrosio Valencia-Romero and P. T. Grogan, The strategy dynamics of collective systems: Underlying hindrances beyond two-actor coordination, PLOS ONE 19(4): e0301394 (S1 Appendix).
Index entries for linear recurrences with constant coefficients, signature (5,-6).

Cf. A000225, A016189, A036561, A097934, A038876, A127071, A127072, A127073, A127074, A002997, A057468, A109235, A281890, A329064, A350771.
a(n) = row sums of A091913, row 2 of A047969, column 1 of A090888 and column 1 of A038719.
Cf. A000392, A240400, A000244, A028243.
Cf. partitions: A241766, A241759.
A diagonal of A262307.

a001047 n = a001047_list !! n
a001047_list = map fst $ iterate (\(u, v) -> (3 * u + v, 2 * v)) (0, 1)
-- Reinhard Zumkeller, Jun 09 2013

[3^n - 2^n: n in [0..30]]; // Vincenzo Librandi, Jul 17 2011

seq(3^n - 2^n, n=0..40); # Giorgio Balzarotti, Nov 18 2006
A001047:=1/(3*z-1)/(2*z-1); # Simon Plouffe in his 1992 dissertation, dropping the initial zero

Table[ 3^n - 2^n, {n, 0, 25} ]
LinearRecurrence[{5, -6}, {0, 1}, 25] (* Harvey P. Dale, Aug 18 2011 *)
Numerator@NestList[(3#+1)/2&,1/2,100] (* Zak Seidov, Oct 03 2011 *)

PARI
```
{a(n) = 3^n - 2^n};
```

[3**n - 2**n for n in range(25)] # Ross La Haye, Aug 19 2005; corrected by David Radcliffe, Jun 26 2016

[lucas_number1(n, 5, 6) for n in range(26)]  # Zerinvary Lajos, Apr 22 2009

G.f.: x/((1-2*x)*(1-3*x)).
a(n) = 5*a(n-1) - 6*a(n-2).
a(n) = 3*a(n-1) + 2^(n-1). - Jon Perry, Aug 23 2002
Starting 0, 0, 1, 5, 19, ... this is 3^n/3 - 2^n/2 + 0^n/6, the binomial transform of A086218. - Paul Barry, Aug 18 2003
a(n) = A083323(n)-1 = A056182(n)/2 = (A002783(n)-1)/2 = (A003063(n+2)-A003063(n+1))/2. - Ralf Stephan, Jan 12 2004
Binomial transform of A000225. - Ross La Haye, Feb 07 2005
a(n) = Sum_{k=0..n-1} binomial(n, k)*2^k. - Ross La Haye, Aug 20 2005
a(n) = 2^(2n) - A083324(n). - Ross La Haye, Sep 10 2005
a(n) = A112626(n, 1). - Ross La Haye, Jan 11 2006
E.g.f.: exp(3*x) - exp(2*x). - Mohammad K. Azarian, Jan 14 2009
a(n) = A217764(n,1). - Ross La Haye, Mar 27 2013
a(n) = 2*a(n-1) + 3^(n-1). - Toby Gottfried, Mar 28 2013
a(n) = A000244(n) - A000079(n). - Omar E. Pol, Mar 28 2013
a(n) = Sum_{k=0..2} Stirling1(2,k)*(k+1)^n = c_2^{(-n)}, poly-Cauchy numbers. - Takao Komatsu, Mar 28 2013
a(n) = A227048(n,A098294(n)). - Reinhard Zumkeller, Jun 30 2013
a(n+1) = Sum_{k=0..n} 2^k*3^(n-k). - J. M. Bergot, Mar 27 2018
Sum_{n>=1} 1/a(n) = A329064. - Amiram Eldar, Nov 20 2020
a(n) = (1/2)*Sum_{k=0..n} binomial(n, k)*(2^(n-k) + 2^k - 2).
a(n) = A001117(n) + 2*A000918(n) + 1. - Ambrosio Valencia-Romero, Mar 08 2022
a(n) = A000225(n) + A028243(n+1). - Ambrosio Valencia-Romero, Mar 09 2022
From Peter Bala, Jun 27 2025: (Start)
exp(Sum_{n >=1} a(2*n)/a(n)*x^n/n) = Sum_{n >= 0} a(n+1)*x^n.
exp(Sum_{n >=1} a(3*n)/a(n)*x^n/n) = 1 + 19*x + 247*x^2 + ... is the g.f. of A019443.
exp(Sum_{n >=1} a(4*n)/a(n)*x^n/n) = 1 + 65*x + 2743*x^2 + ... is the g.f. of A383754.
The following are all examples of telescoping series:
Sum_{n >= 1} 6^n/(a(n)*a(n+1)) = 2, since 6^n/(a(n)*a(n+1)) = b(n) - b(n+1), where b(n) = 2^n/a(n);
Sum_{n >= 1} 18^n/(a(n)*a(n+1)*a(n+2)) = 22/75, since 18^n/(a(n)*a(n+1)*a(n+2)) = c(n) - c(n+1), where c(n) = (5*6^n - 2*4^n)/(15*a(n)*a(n+1));
Sum_{n >= 1} 54^n/(a(n)*a(n+1)*a(n+2)*a(n+3)) = 634/48735 since 54^n/(a(n)*a(n+1)*a(n+2)*a(n+3)) = d(n) - d(n+1), where d(n) = (57*18^n - 38*12^n + 8*8^n)/(513*a(n)*a(n+1)*a(n+2)).
Sum_{n >= 1} 6^n/(a(n)*a(n+2)) = 14/25; Sum_{n >= 1} (-6)^n/(a(n)*a(n+2)) = -6/25.
Sum_{n >= 1} 6^n/(a(n)*a(n+3)) = 306/1805.
Sum_{n >= 1} 6^n/(a(n)*a(n+4)) = 4282/80275; Sum_{n >= 1} (-6)^n/(a(n)*a(n+4)) = -1698/80275. (End)

Edited by Charles R Greathouse IV, Mar 24 2010

A073631 Nonprimes k such that k divides 3^(k-1) - 2^(k-1).

Original entry on oeis.org

1, 65, 133, 529, 793, 1105, 1649, 1729, 2059, 2321, 2465, 2701, 2821, 4187, 5185, 6305, 6541, 6601, 6697, 6817, 7471, 7613, 8113, 8911, 10585, 10963, 11521, 13213, 13333, 13427, 14701, 14981, 15841, 18721, 19171, 19201, 19909, 21349, 21667, 22177, 26065

Offset: 1

Benoit Cloitre, Aug 29 2002

Terms 1,65,2059,6305,19171,... are also in A001047
All primes p>3 divide 3^(p-1) - 2^(p-1). It appears that a(1) = 1 and a(4) = 529 = 23^2 are the only perfect squares in a(n). Most terms of a(n) are squarefree. First 50 nonsquarefree terms of a(n) are the multiples of 23^2. Conjecture: All nonsquarefree terms of a(n) are the multiples of 23^2. Numbers n such that k=n*23^2 divides 3^(k-1) - 2^(k-1) are listed in A130058 = {1, 67, 89, 133, 199, 331, 617, 793, 881, 5281, 8911, 1419, 13333,...}. - Alexander Adamchuk, May 04 2007
Contains all Carmichael numbers (A002997) that are not divisible by 3. - Robert Israel, May 19 2015

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Robert Israel)

Cf. A001047 (3^n - 2^n), A002997.
Cf. A038876, A097934 (primes p such that p divides 3^((p-1)/2) - 2^((p-1)/2)).
Cf. A130059, A130058 (numbers n such that k=n*23^2 divides 3^(k-1) - 2^(k-1)).

[n: n in [1..3*10^4] | not IsPrime(n) and IsDivisibleBy(3^(n-1)-2^(n-1), n)]; // Vincenzo Librandi, May 20 2015

1,op(select(n -> (3 &^ (n-1) - 2 &^ (n-1) mod n = 0 and not isprime(n)), [seq(2*i+1,i=1..10000)])); # Robert Israel, May 19 2015

Select[Range[3 10^4], ! PrimeQ[#] && Mod[3^(# - 1) - 2^(# - 1), #] == 0 &] (* Vincenzo Librandi, May 20 2015 *)
Select[Range[3*10^4], PowerMod[3, # - 1, #] == PowerMod[2, # - 1, #] && !PrimeQ[#] &] (* Amiram Eldar, Mar 27 2021 *)

isok(n) = ! isprime(n) && !((3^(n-1)-2^(n-1)) % n); \\ Michel Marcus, Nov 28 2013

Term 14701 added and more terms from Michel Marcus, Nov 28 2013

cp's OEIS Frontend

A001047 a(n) = 3^n - 2^n.

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A035188 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 6.

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A073631 Nonprimes k such that k divides 3^(k-1) - 2^(k-1).

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A130062 Nonprime numbers k such that k divides 3^((k+1)/2) - 2^((k+1)/2) - 1.

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A130061 Numbers k that divide 3^((k-1)/2) - 2^((k-1)/2) - 1.

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A130063 Primes p such that p divides 3^((p+1)/2) - 2^((p+1)/2) - 1.

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A035188 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = 6.