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%I M2807 N1129 #526 Mar 09 2024 11:44:51
%S 1,3,9,27,81,243,729,2187,6561,19683,59049,177147,531441,1594323,
%T 4782969,14348907,43046721,129140163,387420489,1162261467,3486784401,
%U 10460353203,31381059609,94143178827,282429536481,847288609443,2541865828329,7625597484987
%N Powers of 3: a(n) = 3^n.
%C Same as Pisot sequences E(1, 3), L(1, 3), P(1, 3), T(1, 3). Essentially same as Pisot sequences E(3, 9), L(3, 9), P(3, 9), T(3, 9). See A008776 for definitions of Pisot sequences.
%C Number of (s(0), s(1), ..., s(2n+2)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| = 1 for i = 1, 2, ..., 2n + 2, s(0) = 1, s(2n+2) = 3. - _Herbert Kociemba_, Jun 10 2004
%C a(1) = 1, a(n+1) is the least number such that there are a(n) even numbers between a(n) and a(n+1). Generalization for the sequence of powers of k: 1, k, k^2, k^3, k^4, ... There are a(n) multiples of k-1 between a(n) and a(n+1). - _Amarnath Murthy_, Nov 28 2004
%C a(n) = sum of (n+1)-th row in Triangle A105728. - _Reinhard Zumkeller_, Apr 18 2005
%C With p(n) being the number of integer partitions of n, p(i) being the number of parts of the i-th partition of n, d(i) being the number of different parts of the i-th partition of n, m(i, j) being the multiplicity of the j-th part of the i-th partition of n, Sum_{i = 1..p(n)} being the sum over i and Product_{j = 1..d(i)} being the product over j, one has: a(n) = Sum_{i = 1..p(n)} (p(i)!/(Product_{j = 1..d(i)} m(i, j)!))*2^(p(i) - 1). - _Thomas Wieder_, May 18 2005
%C For any k > 1 in the sequence, k is the first prime power appearing in the prime decomposition of repunit R_k, i.e., of A002275(k). - _Lekraj Beedassy_, Apr 24 2006
%C a(n-1) is the number of compositions of compositions. In general, (k+1)^(n-1) is the number of k-levels nested compositions (e.g., 4^(n-1) is the number of compositions of compositions of compositions, etc.). Each of the n - 1 spaces between elements can be a break for one of the k levels, or not a break at all. - _Franklin T. Adams-Watters_, Dec 06 2006
%C Let S 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), xSy if x is a subset of y. Then a(n) = |S|. - _Ross La Haye_, Dec 22 2006
%C From _Manfred Boergens_, Mar 28 2023: (Start)
%C With regard to the comment by Ross La Haye:
%C Cf. A001047 if either nonempty subsets are considered or x is a proper subset of y.
%C Cf. a(n+1) in A028243 if nonempty subsets are considered and x is a proper subset of y. (End)
%C If X_1, X_2, ..., X_n is a partition of the set {1, 2, ..., 2*n} into blocks of size 2 then, for n >= 1, a(n) is equal to the number of functions f : {1, 2, ..., 2*n} -> {1, 2} such that for fixed y_1, y_2, ..., y_n in {1, 2} we have f(X_i) <> {y_i}, (i = 1, 2, ..., n). - _Milan Janjic_, May 24 2007
%C This is a general comment on all sequences of the form a(n) = [(2^k)-1]^n for all positive integers k. Example 1.1.16 of Stanley's "Enumerative Combinatorics" offers a slightly different version. a(n) in the number of functions f:[n] into P([k]) - {}. a(n) is also the number of functions f:[k] into P([n]) such that the generalized intersection of f(i) for all i in [k] is the empty set. Where [n] = {1, 2, ..., n}, P([n]) is the power set of [n] and {} is the empty set. - _Geoffrey Critzer_, Feb 28 2009
%C a(n) = A064614(A000079(n)) and A064614(m)<a(n) for m < A000079(n). - _Reinhard Zumkeller_, Feb 08 2010
%C 3^(n+1) = (1, 2, 2, 2, ...) dot (1, 1, 3, 9, ..., 3^n); e.g., 3^3 = 27 = (1, 2, 2, 2) dot (1, 1, 3, 9) = (1 + 2 + 6 + 18). - _Gary W. Adamson_, May 17 2010
%C a(n) is the number of generalized compositions of n when there are 3*2^i different types of i, (i = 1, 2, ...). - _Milan Janjic_, Sep 24 2010
%C For n >= 1, a(n-1) is the number of generalized compositions of n when there are 2^(i-1) different types of i, (i = 1, 2, ...). - _Milan Janjic_, Sep 24 2010
%C The sequence in question ("Powers of 3") also describes the number of moves of the k-th disk solving the [RED ; BLUE ; BLUE] or [RED ; RED ; BLUE] pre-colored Magnetic Tower of Hanoi puzzle (cf. A183111 - A183125).
%C a(n) is the number of Stern polynomials of degree n. See A057526. - _T. D. Noe_, Mar 01 2011
%C Positions of records in the number of odd prime factors, A087436. - _Juri-Stepan Gerasimov_, Mar 17 2011
%C Sum of coefficients of the expansion of (1+x+x^2)^n. - _Adi Dani_, Jun 21 2011
%C a(n) is the number of compositions of n elements among {0, 1, 2}; e.g., a(2) = 9 since there are the 9 compositions 0 + 0, 0 + 1, 1 + 0, 0 + 2, 1 + 1, 2 + 0, 1 + 2, 2 + 1, and 2 + 2. [From _Adi Dani_, Jun 21 2011; modified by editors.]
%C Except the first two terms, these are odd numbers n such that no x with 2 <= x <= n - 2 satisfy x^(n-1) == 1 (mod n). - _Arkadiusz Wesolowski_, Jul 03 2011
%C The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 3-colored compositions of n such that no adjacent parts have the same color. - _Milan Janjic_, Nov 17 2011
%C Explanation from _David Applegate_, Feb 20 2017: (Start)
%C Since the preceding comment appears in a large number of sequences, it might be worth adding a proof.
%C The number of compositions of n into exactly k parts is binomial(n-1,k-1).
%C For a p-colored composition of n such that no adjacent parts have the same color, there are exactly p choices for the color of the first part, and p-1 choices for the color of each additional part (any color other than the color of the previous one). So, for a partition into k parts, there are p (p-1)^(k-1) valid colorings.
%C Thus the number of p-colored compositions of n into exactly k parts such that no adjacent parts have the same color is binomial(n-1,k-1) p (p-1)^(k-1).
%C The total number of p-colored compositions of n such that no adjacent parts have the same color is then
%C Sum_{k=1..n} binomial(n-1,k-1) * p * (p-1)^(k-1) = p^n.
%C To see this, note that the binomial expansion of ((p - 1) + 1)^(n - 1) = Sum_{k = 0..n - 1} binomial(n - 1, k) (p - 1)^k 1^(n - 1 - k) = Sum_{k = 1..n} binomial(n - 1, k - 1) (p - 1)^(k - 1).
%C (End)
%C Also, first and least element of the matrix [1, sqrt(2); sqrt(2), 2]^(n+1). - _M. F. Hasler_, Nov 25 2011
%C One-half of the row sums of the triangular version of A035002. - _J. M. Bergot_, Jun 10 2013
%C Form an array with m(0,n) = m(n,0) = 2^n; m(i,j) equals the sum of the terms to the left of m(i,j) and the sum of the terms above m(i,j), which is m(i,j) = Sum_{k=0..j-1} m(i,k) + Sum_{k=0..i-1} m(k,j). The sum of the terms in antidiagonal(n+1) = 4*a(n). - _J. M. Bergot_, Jul 10 2013
%C a(n) = A007051(n+1) - A007051(n), and A007051 are the antidiagonal sums of an array defined by m(0,k) = 1 and m(n,k) = Sum_{c = 0..k - 1} m(n, c) + Sum_{r = 0..n - 1} m(r, k), which is the sum of the terms to left of m(n, k) plus those above m(n, k). m(1, k) = A000079(k); m(2, k) = A045623(k + 1); m(k + 1, k) = A084771(k). - _J. M. Bergot_, Jul 16 2013
%C Define an array to have m(0,k) = 2^k and m(n,k) = Sum_{c = 0..k - 1} m(n, c) + Sum_{r = 0..n - 1} m(r, k), which is the sum of the terms to the left of m(n, k) plus those above m(n, k). Row n = 0 of the array comprises A000079, column k = 0 comprises A011782, row n = 1 comprises A001792. Antidiagonal sums of the array are a(n): 1 = 3^0, 1 + 2 = 3^1, 2 + 3 + 4 = 3^2, 4 + 7 + 8 + 8 = 3^3. - _J. M. Bergot_, Aug 02 2013
%C The sequence with interspersed zeros and o.g.f. x/(1 - 3*x^2), A(2*k) = 0, A(2*k + 1) = 3^k = a(k), k >= 0, can be called hexagon numbers. This is because the algebraic number rho(6) = 2*cos(Pi/6) = sqrt(3) of degree 2, with minimal polynomial C(6, x) = x^2 - 3 (see A187360, n = 6), is the length ratio of the smaller diagonal and the side in the hexagon. Hence rho(6)^n = A(n-1)*1 + A(n)*rho(6), in the power basis of the quadratic number field Q(rho(6)). One needs also A(-1) = 1. See also a Dec 02 2010 comment and the P. Steinbach reference given in A049310. - _Wolfdieter Lang_, Oct 02 2013
%C Numbers k such that sigma(3k) = 3k + sigma(k). - _Jahangeer Kholdi_, Nov 23 2013
%C All powers of 3 are perfect totient numbers (A082897), since phi(3^n) = 2 * 3^(n - 1) for n > 0, and thus Sum_{i = 0..n} phi(3^i) = 3^n. - _Alonso del Arte_, Apr 20 2014
%C The least number k > 0 such that 3^k ends in n consecutive decreasing digits is a 3-term sequence given by {1, 13, 93}. The consecutive increasing digits are {3, 23, 123}. There are 100 different 3-digit endings for 3^k. There are no k-values such that 3^k ends in '012', '234', '345', '456', '567', '678', or '789'. The k-values for which 3^k ends in '123' are given by 93 mod 100. For k = 93 + 100*x, the digit immediately before the run of '123' is {9, 5, 1, 7, 3, 9, 5, 1, 3, 7, ...} for x = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...}, respectively. Thus we see the digit before '123' will never be a 0. So there are no further terms. - _Derek Orr_, Jul 03 2014
%C All elements of A^n where A = (1, 1, 1; 1, 1, 1; 1, 1, 1). - _David Neil McGrath_, Jul 23 2014
%C Counts all walks of length n (open or closed) on the vertices of a triangle containing a loop at each vertex starting from any given vertex. - _David Neil McGrath_, Oct 03 2014
%C a(n) counts walks (closed) on the graph G(1-vertex;1-loop,1-loop,1-loop). - _David Neil McGrath_, Dec 11 2014
%C 2*a(n-2) counts all permutations of a solitary closed walk of length (n) from the vertex of a triangle that contains 2 loops on each of the remaining vertices. In addition, C(m,k)=2*(2^m)*B(m+k-2,m) counts permutations of walks that contain (m) loops and (k) arcs. - _David Neil McGrath_, Dec 11 2014
%C a(n) is the sum of the coefficients of the n-th layer of Pascal's pyramid (a.k.a., Pascal's tetrahedron - see A046816). - _Bob Selcoe_, Apr 02 2016
%C Numbers n such that the trinomial x^(2*n) + x^n + 1 is irreducible over GF(2). Of these only the trinomial for n=1 is primitive. - _Joerg Arndt_, May 16 2016
%C Satisfies Benford's law [Berger-Hill, 2011]. - _N. J. A. Sloane_, Feb 08 2017
%C a(n-1) is also the number of compositions of n if the parts can be runs of any length from 1 to n, and can contain any integers from 1 to n. - _Gregory L. Simay_, May 26 2017
%C Also the number of independent vertex sets and vertex covers in the n-ladder rung graph n P_2. - _Eric W. Weisstein_, Sep 21 2017
%C Also the number of (not necessarily maximal) cliques in the n-cocktail party graph. - _Eric W. Weisstein_, Nov 29 2017
%C a(n-1) is the number of 2-compositions of n; see Hopkins & Ouvry reference. - _Brian Hopkins_, Aug 15 2020
%C a(n) is the number of faces of any dimension (vertices, edges, square faces, etc.) of the n-dimensional hypercube. For example, the 0-dimensional hypercube is a point, and its only face is itself. The 1-dimensional hypercube is a line, which has two vertices and an edge. The 2-dimensional hypercube is a square, which has four vertices, four edges, and a square face. - _Kevin Long_, Mar 14 2023
%C Number of pairs (A,B) of subsets of M={1,2,...,n} with union(A,B)=M. For nonempty subsets cf. A058481. - _Manfred Boergens_, Mar 28 2023
%C From _Jianing Song_, Sep 27 2023: (Start)
%C a(n) is the number of disjunctive clauses of n variables up to equivalence. A disjunctive clause is a propositional formula of the form l_1 OR ... OR l_m, where l_1, ..., l_m are distinct elements in {x_1, ..., x_n, NOT x_1, ..., NOT x_n} for n variables x_1, ... x_n, and no x_i and NOT x_i appear at the same time. For each 1 <= i <= n, we can have neither of x_i or NOT x_i, only x_i or only NOT x_i appearing in a disjunctive clause, so the number of such clauses is 3^n. Viewing the propositional formulas of n variables as functions {0,1}^n -> {0,1}, a disjunctive clause corresponds to a function f such that the inverse image of 0 is of the form A_1 X ... X A_n, where A_i is nonempty for all 1 <= i <= n. Since each A_i has 3 choices ({0}, {1} or {0,1}), we also find that the number of disjunctive clauses of n variables is 3^n.
%C Equivalently, a(n) is the number of conjunctive clauses of n variables. (End)
%C The finite subsequence a(2), a(3), a(4), a(5) = 9, 27, 81, 243 is one of only two geometric sequences that can be formed with all interior angles (all integer, in degrees) of a simple polygon. The other sequence is a subsequence of A007283 (see comment there). - _Felix Huber_, Feb 15 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).
%D Doron Zeilberger, The Amazing 3^n Theorem and its even more Amazing Proof [Discovered by Xavier G. Viennot and his École Bordelaise gang], arXiv:1208.2258, 2012.
%H T. D. Noe, <a href="/A000244/b000244.txt">Table of n, a(n) for n = 0..200</a>
%H T. Banchoff, <a href="https://www.math.brown.edu/tbanchof/Beyond3d/chapter4/section07.html">Counting the Faces of Higher-Dimensional Cubes</a>, Beyond the Third Dimension: Geometry, computer graphics and higher dimensions, Scientific American Library, 1996.
%H Arno Berger and Theodore P. Hill, <a href="http://dx.doi.org/10.1007/s00283-010-9182-3">Benford's law strikes back: no simple explanation in sight for mathematical gem</a>, The Mathematical Intelligencer 33.1 (2011): 85-91.
%H A. Bostan, <a href="https://www-apr.lip6.fr/sem-comb-slides/IHP-bostan.pdf">Computer Algebra for Lattice Path Combinatorics</a>, Séminaire de Combinatoire Ph. Flajolet, Mar 28 2013.
%H Peter J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H F. Javier de Vega, <a href="https://arxiv.org/abs/2003.13378">An extension of Furstenberg's theorem of the infinitude of primes</a>, arXiv:2003.13378 [math.NT], 2020.
%H Nachum Dershowitz, <a href="https://arxiv.org/abs/2006.06516">Between Broadway and the Hudson: A Bijection of Corridor Paths</a>, arXiv:2006.06516 [math.CO], 2020.
%H Joël Gay and Vincent Pilaud, <a href="https://arxiv.org/abs/1804.06572">The weak order on Weyl posets</a>, arXiv:1804.06572 [math.CO], 2018.
%H Brian Hopkins and Stéphane Ouvry, <a href="https://arxiv.org/abs/2008.04937">Combinatorics of Multicompositions</a>, arXiv:2008.04937 [math.CO], 2020.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=7">Encyclopedia of Combinatorial Structures 7</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=268">Encyclopedia of Combinatorial Structures 268</a>
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulae for Some Functions on Finite Sets</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%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 László Németh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Nemeth/nemeth6.html">The trinomial transform triangle</a>, J. Int. Seqs., Vol. 21 (2018), Article 18.7.3. Also <a href="https://arxiv.org/abs/1807.07109">arXiv:1807.07109</a> [math.NT], 2018.
%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 Yash Puri and Thomas Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/WARD/short.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Clique.html">Clique</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CocktailPartyGraph.html">Cocktail Party Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HanoiGraph.html">Hanoi Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IndependentVertexSet.html">Independent Vertex Set</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LadderRungGraph.html">Ladder Rung Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SierpinskiSieveGraph.html">Sierpiński Sieve Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/VertexCover.html">Vertex Cover</a>
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (3).
%H <a href="/index/Be#Benford">Index entries for sequences related to Benford's law</a>
%F a(n) = 3^n.
%F a(0) = 1; a(n) = 3*a(n-1).
%F G.f.: 1/(1-3*x).
%F E.g.f.: exp(3*x).
%F a(n) = n!*Sum_{i + j + k = n, i, j, k >= 0} 1/(i!*j!*k!). - _Benoit Cloitre_, Nov 01 2002
%F a(n) = Sum_{k = 0..n} 2^k*binomial(n, k), binomial transform of A000079.
%F a(n) = A090888(n, 2). - _Ross La Haye_, Sep 21 2004
%F a(n) = 2^(2n) - A005061(n). - _Ross La Haye_, Sep 10 2005
%F a(n) = A112626(n, 0). - _Ross La Haye_, Jan 11 2006
%F Hankel transform of A007854. - _Philippe Deléham_, Nov 26 2006
%F a(n) = 2*StirlingS2(n+1,3) + StirlingS2(n+2,2) = 2*(StirlingS2(n+1,3) + StirlingS2(n+1,2)) + 1. - _Ross La Haye_, Jun 26 2008
%F a(n) = 2*StirlingS2(n+1, 3) + StirlingS2(n+2, 2) = 2*(StirlingS2(n+1, 3) + StirlingS2(n+1, 2)) + 1. - _Ross La Haye_, Jun 09 2008
%F Sum_{n >= 0} 1/a(n) = 3/2. - _Gary W. Adamson_, Aug 29 2008
%F If p(i) = Fibonacci(2i-2) and if A is the Hessenberg matrix of order n defined by A(i, j) = p(j-i+1), (i <= j), A(i, j) = -1, (i = j+1), and A(i, j) = 0 otherwise, then, for n >= 1, a(n-1) = det A. - _Milan Janjic_, May 08 2010
%F G.f. A(x) = M(x)/(1-M(x))^2, M(x) - o.g.f for Motzkin numbers (A001006). - _Vladimir Kruchinin_, Aug 18 2010
%F a(n) = A133494(n+1). - _Arkadiusz Wesolowski_, Jul 27 2011
%F 2/3 + 3/3^2 + 2/3^3 + 3/3^4 + 2/3^5 + ... = 9/8. [Jolley, Summation of Series, Dover, 1961]
%F a(n) = Sum_{k=0..n} A207543(n,k)*4^(n-k). - _Philippe Deléham_, Feb 25 2012
%F a(n) = Sum_{k=0..n} A125185(n,k). - _Philippe Deléham_, Feb 26 2012
%F Sum_{n > 0} Mobius(n)/a(n) = 0.181995386702633887827... (see A238271). - _Alonso del Arte_, Aug 09 2012. See also the sodium 3s orbital energy in table V of J. Chem. Phys. 53 (1970) 348.
%F a(n) = (tan(Pi/3))^(2*n). - _Bernard Schott_, May 06 2022
%F a(n-1) = binomial(2*n-1, n) + Sum_{k >= 1} binomial(2*n, n+3*k)*(-1)^k. - _Greg Dresden_, Oct 14 2022
%F G.f.: Sum_{k >= 0} x^k/(1-2*x)^(k+1). - _Kevin Long_, Mar 14 2023
%e G.f. = 1 + 3*x + 9*x^2 + 27*x^3 + 81*x^4 + 243*x^5 + 729*x^6 + 2187*x^7 + ...
%p A000244 := n->3^n; [ seq(3^n, n=0..50) ];
%p A000244:=-1/(-1+3*z); # _Simon Plouffe_ in his 1992 dissertation.
%t Table[3^n, {n, 0, 30}] (* _Stefan Steinerberger_, Apr 01 2006 *)
%t 3^Range[0, 30] (* _Wesley Ivan Hurt_, Jul 04 2014 *)
%t LinearRecurrence[{3}, {1}, 20] (* _Eric W. Weisstein_, Sep 21 2017 *)
%t CoefficientList[Series[1/(1 - 3 x), {x, 0, 20}], x] (* _Eric W. Weisstein_, Sep 21 2017 *)
%t NestList[3#&,1,30] (* _Harvey P. Dale_, Feb 20 2020 *)
%o (PARI) A000244(n) = 3^n \\ _Michael B. Porter_, Nov 03 2009
%o (Haskell)
%o a000244 = (3 ^) -- _Reinhard Zumkeller_, Nov 14 2011
%o a000244_list = iterate (* 3) 1 -- _Reinhard Zumkeller_, Apr 04 2012
%o (Maxima) makelist(3^n, n, 0, 30); /* _Martin Ettl_, Nov 05 2012 */
%o (Magma) [ 3^n : n in [0..30] ]; // _Wesley Ivan Hurt_, Jul 04 2014
%o (Scala) val powersOf3: LazyList[BigInt] = LazyList.iterate(1: BigInt)(_ * 3)
%o (0 to 26).map(powersOf3(_)) // _Alonso del Arte_, May 03 2020
%o (Python)
%o def A000244(n): return 3**n # _Chai Wah Wu_, Nov 10 2022
%Y Cf. A008776 (2*a(n), and first differences).
%Y a(n) = A092477(n, 2) for n > 0.
%Y a(n) = A159991(n) / A009964(n).
%Y Cf. A100772, A035002. Row sums of A125076 and A153279.
%Y a(n) = A217764(0, n).
%Y Cf. A046816, A006521, A014945, A275414 (multisets).
%Y The following are parallel families: A000079 (2^n), A004094 (2^n reversed), A028909 (2^n sorted up), A028910 (2^n sorted down), A036447 (double and reverse), A057615 (double and sort up), A263451 (double and sort down); A000244 (3^n), A004167 (3^n reversed), A321540 (3^n sorted up), A321539 (3^n sorted down), A163632 (triple and reverse), A321542 (triple and sort up), A321541 (triple and sort down).
%Y Cf. A001047, A028243, A058481.
%K nonn,nice,easy,core
%O 0,2
%A _N. J. A. Sloane_
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