OFFSET
0,3
COMMENTS
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
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)
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
REFERENCES
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).
LINKS
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.
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).
FORMULA
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) = Sum_{k=0..2} Stirling1(2,k)*(k+1)^n = c_2^{(-n)}, poly-Cauchy numbers. - Takao Komatsu, Mar 28 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).
MAPLE
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
MATHEMATICA
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 *)
PROG
(Python) [3**n - 2**n for n in range(25)] # Ross La Haye, Aug 19 2005; corrected by David Radcliffe, Jun 26 2016
(Sage) [lucas_number1(n, 5, 6) for n in range(26)] # Zerinvary Lajos, Apr 22 2009
(PARI) {a(n) = 3^n - 2^n};
(Magma) [3^n - 2^n: n in [0..30]]; // Vincenzo Librandi, Jul 17 2011
(Haskell)
a001047 n = a001047_list !! n
a001047_list = map fst $ iterate (\(u, v) -> (3 * u + v, 2 * v)) (0, 1)
-- Reinhard Zumkeller, Jun 09 2013
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
Edited by Charles R Greathouse IV, Mar 24 2010
STATUS
approved