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%I M3887 N1596 #314 Apr 24 2024 13:08:29
%S 0,1,5,19,65,211,665,2059,6305,19171,58025,175099,527345,1586131,
%T 4766585,14316139,42981185,129009091,387158345,1161737179,3485735825,
%U 10458256051,31376865305,94134790219,282412759265,847255055011,2541798719465,7625463267259,22876524019505
%N a(n) = 3^n - 2^n.
%C a(n+1) is the sum of the elements in the n-th row of triangle pertaining to A036561. - _Amarnath Murthy_, Jan 02 2002
%C 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
%C With offset 1, partial sums of A027649. - _Paul Barry_, Jun 24 2003
%C 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
%C 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
%C a(n+1) is the sum of n-th row of A036561. - _Reinhard Zumkeller_, May 14 2006
%C 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
%C 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
%C From _Alexander Adamchuk_, Jan 04 2007: (Start)
%C a(n) is prime for n in A057468.
%C p divides a(p) - 1 for prime p.
%C Quotients (3^p - 2^p - 1)/p, where p = prime(n), are listed in A127071.
%C Numbers k such that k divides 3^k - 2^k - 1 are listed in A127072.
%C 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.
%C Numbers n such that n^2 divides 3^n - 2^n - 1 are listed in A127074.
%C 5 divides a(2n).
%C 5^2 divides a(2*5n).
%C 5^3 divides a(2*5^2n).
%C 5^4 divides a(2*5^3n).
%C 7^2 divides a(6*7n).
%C 13 divides a(4n).
%C 13^2 divides a(4*13n).
%C 19 divides a(3n).
%C 19^2 divides a(3*19n).
%C 23^2 divides a(11n).
%C 23^3 divides a(11*23n).
%C 23^4 divides a(11*23^2n).
%C 29 divides a(7n).
%C p divides a((p-1)n) for prime p>3.
%C p divides a((p-1)/2)) for prime p in A097936. Also primes p such that 6 is a square mod p, except {2,3}, A038876(n).
%C p^(k+1) divides a(p^k*(p-1)/2*n) for prime p in A097936.
%C p^(k+1) divides a(p^k*(p-1)*n) for prime p>3.
%C Note the exception that for p = 23, p^(k+2) divides a(p^k*(p-1)/2*n).
%C There are no more such exceptions for primes p up to 600000. (End)
%C a(n) divides a(q*(n+1)-1), for all q integer. _Leonardo Sarasua_, Apr 15 2024
%C Final digits of terms follow sequence 1,5,9,5. - _Enoch Haga_, Nov 26 2007
%C 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
%C Partial sums give A000392. - _Jon Perry_, Apr 05 2014
%C 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
%C 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
%C 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
%C 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
%C 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
%C From _Manfred Boergens_, Mar 29 2023: (Start)
%C With regard to the comments by Ross La Haye and Jianing Song: Omitting "proper" gives A000244.
%C 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)
%C a(n) is the number of n-digit numbers whose smallest decimal digit is 7. - _Stefano Spezia_, Nov 15 2023
%C 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
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A001047/b001047.txt">Table of n, a(n) for n=0..200</a>
%H A. Abdurrahman, <a href="https://arxiv.org/abs/1909.10889">CM Method and Expansion of Numbers</a>, arXiv:1909.10889 [math.NT], 2019.
%H Nathan Bliss, Ben Fulan, Stephen Lovett and Jeff Sommars, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.120.06.519">Strong divisibility, cyclotomic polynomials and iterated polynomials</a>, Am. Math. Monthly, Vol. 120, No. 6 (2013), pp. 519-536.
%H John Elias, <a href="/A001047/a001047.png">Illustration: Sierpinski half-hexagons</a>, <a href="/A001047/a001047_1.png">Illustration: Nicomachus triangle 2^n & 3^n correlation</a>, <a href="/A001047/a001047_2.png">Koch Snowflake Fractal Configuration</a>.
%H Joël Gay, <a href="https://tel.archives-ouvertes.fr/tel-01861199">Representation of Monoids and Lattice Structures in the Combinatorics of Weyl Groups</a>, Doctoral Thesis, Discrete Mathematics [cs.DM], Université Paris-Saclay, 2018.
%H Samuele Giraudo, <a href="http://doi.org/10.1007/s10801-014-0543-4">Combinatorial operads from monoids</a>, Journal of Algebraic Combinatorics, Vol. 41, No. 2 (2015), pp. 493-538; <a href="http://arxiv.org/abs/1306.6938">arXiv preprint</a>, arXiv preprint arXiv:1306.6938 [math.CO], 2013-2015.
%H Samuele Giraudo, <a href="https://doi.org/10.1016/j.aam.2016.02.003">Pluriassociative algebras I: The pluriassociative operad</a>, Advances in Applied Mathematics, Vol. 77 (2016), pp. 1-42; <a href="https://arxiv.org/abs/1603.01040">arXiv preprint</a>, arXiv:1603.01040 [math.CO], 2016.
%H Richard K. Guy, <a href="/A002186/a002186.pdf">Letters to N. J. A. Sloane, June-August 1968</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=397">Encyclopedia of Combinatorial Structures 397</a>.
%H B. D. Josephson and J. M. Boardman, <a href="https://archive.org/details/eureka-24/page/19/mode/2up">Problems Drive 1961</a>, Eureka, The Journal of the Archimedeans, Vol. 24 (1961), p. 20; <a href="https://www.archim.org.uk/eureka/archive/Eureka-24.pdf">entire volume</a>.
%H Germain Kreweras, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k480296q/f583.image">Inversion des polynômes de Bell bidimensionnels et application au dénombrement des relations binaires connexes</a>, C. R. Acad. Sci. Paris Ser. A-B, Vol. 268 (1969), pp. A577-A579.
%H Ross La Haye, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/LaHaye/lahaye5.html">Binary Relations on the Power Set of an n-Element Set</a>, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
%H Richard Miles, <a href="https://doi.org/10.1090/S0002-9947-2013-05829-1">Synchronization points and associated dynamical invariants</a>, Trans. Amer. Math. Soc., Vol. 365, No. 10 (2013), pp. 5503-5524.
%H Rajesh Kumar Mohapatra and Tzung-Pei Hong, <a href="https://doi.org/10.3390/math10071161">On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences</a>, Mathematics (2022) Vol. 10, No. 7, 1161.
%H Jon Perry, <a href="https://web.archive.org/web/20060923015735/http://www.users.globalnet.co.uk/~perry/maths/collatz/collatz.htm">Relation to Collatz problem</a>.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H Kalika Prasad, Munesh Kumari, Rabiranjan Mohanta, and Hrishikesh Mahato, <a href="https://arxiv.org/abs/2307.08073">The sequence of higher order Mersenne numbers and associated binomial transforms</a>, arXiv:2307.08073 [math.NT], 2023.
%H Ambrosio Valencia-Romero and P. T. Grogan, <a href="https://doi.org/10.1371/journal.pone.0301394">The strategy dynamics of collective systems: Underlying hindrances beyond two-actor coordination</a>, PLOS ONE 19(4): e0301394 (<a href="https://doi.org/10.1371/journal.pone.0301394.s001">S1 Appendix</a>).
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-6).
%F G.f.: x/((1-2*x)*(1-3*x)).
%F a(n) = 5*a(n-1) - 6*a(n-2).
%F a(n) = 3*a(n-1) + 2^(n-1). - _Jon Perry_, Aug 23 2002
%F 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
%F a(n) = A083323(n)-1 = A056182(n)/2 = (A002783(n)-1)/2 = (A003063(n+2)-A003063(n+1))/2. - _Ralf Stephan_, Jan 12 2004
%F Binomial transform of A000225. - _Ross La Haye_, Feb 07 2005
%F a(n) = Sum_{k=0..n-1} binomial(n, k)*2^k. - _Ross La Haye_, Aug 20 2005
%F a(n) = 2^(2n) - A083324(n). - _Ross La Haye_, Sep 10 2005
%F a(n) = A112626(n, 1). - _Ross La Haye_, Jan 11 2006
%F E.g.f.: exp(3*x) - exp(2*x). - _Mohammad K. Azarian_, Jan 14 2009
%F a(n) = A217764(n,1). - _Ross La Haye_, Mar 27 2013
%F a(n) = 2*a(n-1) + 3^(n-1). - _Toby Gottfried_, Mar 28 2013
%F a(n) = A000244(n) - A000079(n). - _Omar E. Pol_, Mar 28 2013
%F a(n) = Sum_{k=0..2} Stirling1(2,k)*(k+1)^n = c_2^{(-n)}, poly-Cauchy numbers. - _Takao Komatsu_, Mar 28 2013
%F a(n) = A227048(n,A098294(n)). - _Reinhard Zumkeller_, Jun 30 2013
%F a(n+1) = Sum_{k=0..n} 2^k*3^(n-k). - _J. M. Bergot_, Mar 27 2018
%F Sum_{n>=1} 1/a(n) = A329064. - _Amiram Eldar_, Nov 20 2020
%F a(n) = (1/2)*Sum_{k=0..n} binomial(n, k)*(2^(n-k) + 2^k - 2).
%F a(n) = A001117(n) + 2*A000918(n) + 1. - _Ambrosio Valencia-Romero_, Mar 08 2022
%F a(n) = A000225(n) + A028243(n+1). - _Ambrosio Valencia-Romero_, Mar 09 2022
%p seq(3^n - 2^n, n=0..40); # _Giorgio Balzarotti_, Nov 18 2006
%p A001047:=1/(3*z-1)/(2*z-1); # _Simon Plouffe_ in his 1992 dissertation, dropping the initial zero
%t Table[ 3^n - 2^n, {n, 0, 25} ]
%t LinearRecurrence[{5, -6}, {0, 1}, 25] (* _Harvey P. Dale_, Aug 18 2011 *)
%t Numerator@NestList[(3#+1)/2&,1/2,100] (* _Zak Seidov_, Oct 03 2011 *)
%o (Python) [3**n - 2**n for n in range(25)] # _Ross La Haye_, Aug 19 2005; corrected by _David Radcliffe_, Jun 26 2016
%o (Sage) [lucas_number1(n, 5, 6) for n in range(26)] # _Zerinvary Lajos_, Apr 22 2009
%o (PARI) {a(n) = 3^n - 2^n};
%o (Magma) [3^n - 2^n: n in [0..30]]; // _Vincenzo Librandi_, Jul 17 2011
%o (Haskell)
%o a001047 n = a001047_list !! n
%o a001047_list = map fst $ iterate (\(u, v) -> (3 * u + v, 2 * v)) (0, 1)
%o -- _Reinhard Zumkeller_, Jun 09 2013
%Y Cf. A000225, A016189, A036561, A097936, A038876, A127071, A127072, A127073, A127074, A002997, A057468, A109235, A281890, A329064, A350771.
%Y a(n) = row sums of A091913, row 2 of A047969, column 1 of A090888 and column 1 of A038719.
%Y Cf. A000392, A240400, A000244, A028243.
%Y Cf. partitions: A241766, A241759.
%Y A diagonal of A262307.
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_, _R. K. Guy_
%E Edited by _Charles R Greathouse IV_, Mar 24 2010
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