A183111 -id:A183111 - cp's OEIS frontend

A000244 Powers of 3: a(n) = 3^n.

1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907, 43046721, 129140163, 387420489, 1162261467, 3486784401, 10460353203, 31381059609, 94143178827, 282429536481, 847288609443, 2541865828329, 7625597484987

Offset: 0

N. J. A. Sloane

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.
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
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
a(n) = sum of (n+1)-th row in Triangle A105728. - Reinhard Zumkeller, Apr 18 2005
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
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
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
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
From Manfred Boergens, Mar 28 2023: (Start)
With regard to the comment by Ross La Haye:
Cf. A001047 if either nonempty subsets are considered or x is a proper subset of y.
Cf. a(n+1) in A028243 if nonempty subsets are considered and x is a proper subset of y. (End)
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
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
a(n) = A064614(A000079(n)) and A064614(m)A000079(n). - Reinhard Zumkeller, Feb 08 2010
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
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
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
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).
a(n) is the number of Stern polynomials of degree n. See A057526. - T. D. Noe, Mar 01 2011
Positions of records in the number of odd prime factors, A087436. - Juri-Stepan Gerasimov, Mar 17 2011
Sum of coefficients of the expansion of (1+x+x^2)^n. - Adi Dani, Jun 21 2011
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.]
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
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
Explanation from David Applegate, Feb 20 2017: (Start)
Since the preceding comment appears in a large number of sequences, it might be worth adding a proof.
The number of compositions of n into exactly k parts is binomial(n-1,k-1).
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.
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).
The total number of p-colored compositions of n such that no adjacent parts have the same color is then
Sum_{k=1..n} binomial(n-1,k-1) * p * (p-1)^(k-1) = p^n.
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).
(End)
Also, first and least element of the matrix [1, sqrt(2); sqrt(2), 2]^(n+1). - M. F. Hasler, Nov 25 2011
One-half of the row sums of the triangular version of A035002. - J. M. Bergot, Jun 10 2013
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
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
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
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
Numbers k such that sigma(3k) = 3k + sigma(k). - Jahangeer Kholdi, Nov 23 2013
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
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
All elements of A^n where A = (1, 1, 1; 1, 1, 1; 1, 1, 1). - David Neil McGrath, Jul 23 2014
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
a(n) counts walks (closed) on the graph G(1-vertex;1-loop,1-loop,1-loop). - David Neil McGrath, Dec 11 2014
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
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
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
Satisfies Benford's law [Berger-Hill, 2011]. - N. J. A. Sloane, Feb 08 2017
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
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
Also the number of (not necessarily maximal) cliques in the n-cocktail party graph. - Eric W. Weisstein, Nov 29 2017
a(n-1) is the number of 2-compositions of n; see Hopkins & Ouvry reference. - Brian Hopkins, Aug 15 2020
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
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
From Jianing Song, Sep 27 2023: (Start)
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.
Equivalently, a(n) is the number of conjunctive clauses of n variables. (End)
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

			G.f. = 1 + 3*x + 9*x^2 + 27*x^3 + 81*x^4 + 243*x^5 + 729*x^6 + 2187*x^7 + ...

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
T. Banchoff, Counting the Faces of Higher-Dimensional Cubes, Beyond the Third Dimension: Geometry, computer graphics and higher dimensions, Scientific American Library, 1996.
Arno Berger and Theodore P. Hill, Benford's law strikes back: no simple explanation in sight for mathematical gem, The Mathematical Intelligencer 33.1 (2011): 85-91.
A. Bostan, Computer Algebra for Lattice Path Combinatorics, Séminaire de Combinatoire Ph. Flajolet, Mar 28 2013.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.
Nachum Dershowitz, Between Broadway and the Hudson: A Bijection of Corridor Paths, arXiv:2006.06516 [math.CO], 2020.
Joël Gay and Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.
Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 7
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 268
Milan Janjic, Enumerative Formulae for Some Functions on Finite Sets
Tanya Khovanova, Recursive Sequences
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.
László Németh, The trinomial transform triangle, J. Int. Seqs., Vol. 21 (2018), Article 18.7.3. Also arXiv:1807.07109 [math.NT], 2018.
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
Yash Puri and Thomas Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Eric Weisstein's World of Mathematics, Clique.
Eric Weisstein's World of Mathematics, Cocktail Party Graph.
Eric Weisstein's World of Mathematics, Hanoi Graph.
Eric Weisstein's World of Mathematics, Independent Vertex Set.
Eric Weisstein's World of Mathematics, Ladder Rung Graph.
Eric Weisstein's World of Mathematics, Sierpiński Gasket Graph.
Eric Weisstein's World of Mathematics, Vertex Cover.
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.
Index entries for "core" sequences
Index entries for related partition-counting sequences
Index entries for linear recurrences with constant coefficients, signature (3).
Index entries for sequences related to Benford's law

Cf. A008776 (2*a(n), and first differences).
a(n) = A092477(n, 2) for n > 0.
a(n) = A159991(n) / A009964(n).
Cf. A100772, A035002. Row sums of A125076 and A153279.
a(n) = A217764(0, n).
Cf. A046816, A006521, A014945, A275414 (multisets).
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).
Cf. A001047, A028243, A058481.

a000244 = (3 ^)  -- Reinhard Zumkeller, Nov 14 2011
a000244_list = iterate (* 3) 1  -- Reinhard Zumkeller, Apr 04 2012

[ 3^n : n in [0..30] ]; // Wesley Ivan Hurt, Jul 04 2014

A000244 := n->3^n; [ seq(3^n, n=0..50) ];
A000244:=-1/(-1+3*z); # Simon Plouffe in his 1992 dissertation.

Table[3^n, {n, 0, 30}] (* Stefan Steinerberger, Apr 01 2006 *)
3^Range[0, 30] (* Wesley Ivan Hurt, Jul 04 2014 *)
LinearRecurrence[{3}, {1}, 20] (* Eric W. Weisstein, Sep 21 2017 *)
CoefficientList[Series[1/(1 - 3 x), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 21 2017 *)
NestList[3#&,1,30] (* Harvey P. Dale, Feb 20 2020 *)

makelist(3^n, n, 0, 30); /* Martin Ettl, Nov 05 2012 */

A000244(n) = 3^n \\ Michael B. Porter, Nov 03 2009

def A000244(n): return 3**n # Chai Wah Wu, Nov 10 2022

val powersOf3: LazyList[BigInt] = LazyList.iterate(1: BigInt)(_ * 3)
(0 to 26).map(powersOf3()) // _Alonso del Arte, May 03 2020

a(n) = 3^n.
a(0) = 1; a(n) = 3*a(n-1).
G.f.: 1/(1-3*x).
E.g.f.: exp(3*x).
a(n) = n!*Sum_{i + j + k = n, i, j, k >= 0} 1/(i!*j!*k!). - Benoit Cloitre, Nov 01 2002
a(n) = Sum_{k = 0..n} 2^k*binomial(n, k), binomial transform of A000079.
a(n) = A090888(n, 2). - Ross La Haye, Sep 21 2004
a(n) = 2^(2n) - A005061(n). - Ross La Haye, Sep 10 2005
a(n) = A112626(n, 0). - Ross La Haye, Jan 11 2006
Hankel transform of A007854. - Philippe Deléham, Nov 26 2006
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
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
Sum_{n >= 0} 1/a(n) = 3/2. - Gary W. Adamson, Aug 29 2008
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
G.f. A(x) = M(x)/(1-M(x))^2, M(x) - o.g.f for Motzkin numbers (A001006). - Vladimir Kruchinin, Aug 18 2010
a(n) = A133494(n+1). - Arkadiusz Wesolowski, Jul 27 2011
2/3 + 3/3^2 + 2/3^3 + 3/3^4 + 2/3^5 + ... = 9/8. [Jolley, Summation of Series, Dover, 1961]
a(n) = Sum_{k=0..n} A207543(n,k)*4^(n-k). - Philippe Deléham, Feb 25 2012
a(n) = Sum_{k=0..n} A125185(n,k). - Philippe Deléham, Feb 26 2012
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.
a(n) = (tan(Pi/3))^(2*n). - Bernard Schott, May 06 2022
a(n-1) = binomial(2*n-1, n) + Sum_{k >= 1} binomial(2*n, n+3*k)*(-1)^k. - Greg Dresden, Oct 14 2022
G.f.: Sum_{k >= 0} x^k/(1-2*x)^(k+1). - Kevin Long, Mar 14 2023

A003462 a(n) = (3^n - 1)/2.

Original entry on oeis.org

0, 1, 4, 13, 40, 121, 364, 1093, 3280, 9841, 29524, 88573, 265720, 797161, 2391484, 7174453, 21523360, 64570081, 193710244, 581130733, 1743392200, 5230176601, 15690529804, 47071589413, 141214768240, 423644304721, 1270932914164

Offset: 0

N. J. A. Sloane

Partial sums of A000244. Values of base 3 strings of 1's.
a(n) = (3^n-1)/2 is also the number of different nonparallel lines determined by pair of vertices in the n dimensional hypercube. Example: when n = 2 the square has 4 vertices and then the relevant lines are: x = 0, y = 0, x = 1, y = 1, y = x, y = 1-x and when we identify parallel lines only 4 remain: x = 0, y = 0, y = x, y = 1 - x so a(2) = 4. - Noam Katz (noamkj(AT)hotmail.com), Feb 11 2001
Also number of 3-block bicoverings of an n-set (if offset is 1, cf. A059443). - Vladeta Jovovic, Feb 14 2001
3^a(n) is the highest power of 3 dividing (3^n)!. - Benoit Cloitre, Feb 04 2002
Apart from the a(0) and a(1) terms, maximum number of coins among which a lighter or heavier counterfeit coin can be identified (but not necessarily labeled as heavier or lighter) by n weighings. - Tom Verhoeff, Jun 22 2002, updated Mar 23 2017
n such that A001764(n) is not divisible by 3. - Benoit Cloitre, Jan 14 2003
Consider the mapping f(a/b) = (a + 2b)/(2a + b). Taking a = 1, b = 2 to start with and carrying out this mapping repeatedly on each new (reduced) rational number gives the sequence 1/2, 4/5, 13/14, 40/41, ... converging to 1. Sequence contains the numerators = (3^n-1)/2. The same mapping for N, i.e., f(a/b) = (a + Nb)/(a+b) gives fractions converging to N^(1/2). - Amarnath Murthy, Mar 22 2003
Binomial transform of A000079 (with leading zero). - Paul Barry, Apr 11 2003
With leading zero, inverse binomial transform of A006095. - Paul Barry, Aug 19 2003
Number of walks of length 2*n + 2 in the path graph P_5 from one end to the other one. Example: a(2) = 4 because in the path ABCDE we have ABABCDE, ABCBCDE, ABCDCDE and ABCDEDE. - Emeric Deutsch, Apr 02 2004
The number of triangles of all sizes (not counting holes) in Sierpiński's triangle after n inscriptions. - Lee Reeves (leereeves(AT)fastmail.fm), May 10 2004
Number of (s(0), s(1), ..., s(2n+1)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| = 1 for i = 1, 2, ..., 2*n + 1, s(0) = 1, s(2n+1) = 4. - Herbert Kociemba, Jun 10 2004
Number of non-degenerate right-angled incongruent integer-edged Heron triangles whose circumdiameter is the product of n distinct primes of shape 4k + 1. - Alex Fink and R. K. Guy, Aug 18 2005
Also numerator of the sum of the reciprocals of the first n powers of 3, with A000244 being the sequence of denominators. With the exception of n < 2, the base 10 digital root of a(n) is always 4. In base 3 the digital root of a(n) is the same as the digital root of n. - Alonso del Arte, Jan 24 2006
The sequence 3*a(n), n >= 1, gives the number of edges of the Hanoi graph H_3^{n} with 3 pegs and n >= 1 discs. - Daniele Parisse, Jul 28 2006
Numbers n such that a(n) is prime are listed in A028491 = {3, 7, 13, 71, 103, 541, 1091, ...}. 2^(m+1) divides a(2^m*k) for m > 0. 5 divides a(4k). 5^2 divides a(20k). 7 divides a(6k). 7^2 divides a(42k). 11^2 divides a(5k). 13 divides a(3k). 17 divides a(16k). 19 divides a(18k). 1093 divides a(7k). 41 divides a(8k). p divides a((p-1)/5) for prime p = {41, 431, 491, 661, 761, 1021, 1051, 1091, 1171, ...}. p divides a((p-1)/4) for prime p = {13, 109, 181, 193, 229, 277, 313, 421, 433, 541, ...}. p divides a((p-1)/3) for prime p = {61, 67, 73, 103, 151, 193, 271, 307, 367, ...} = A014753, 3 and -3 are both cubes (one implies other) mod these primes p = 1 mod 6. p divides a((p-1)/2) for prime p = {11, 13, 23, 37, 47, 59, 61, 71, 73, 83, 97, ...} = A097933(n). p divides a(p-1) for prime p > 7. p^2 divides a(p*(p-1)k) for all prime p except p = 3. p^3 divides a(p*(p-1)*(p-2)k) for prime p = 11. - Alexander Adamchuk, Jan 22 2007
Let P(A) be the power set of an n-element set A. Then a(n) = the number of [unordered] pairs of elements {x,y} of P(A) for which x and y are disjoint [and both nonempty]. Wieder calls these "disjoint usual 2-combinations". - Ross La Haye, Jan 10 2008 [This is because each of the elements of {1, 2, ..., n} can be in the first subset, in the second or in neither. Because there are three options for each, the total number of options is 3^n. However, since the sets being empty is not an option we subtract 1 and since the subsets are unordered we then divide by 2! (The number of ways two objects can be arranged.) Thus we obtain (3^n-1)/2 = a(n). - Chayim Lowen, Mar 03 2015]
Also, still with P(A) being the power set of a n-element set A, a(n) is the number of 2-element subsets {x,y} of P(A) such that the union of x and y is equal to A. Cf. A341590. - Fabio Visonà, Feb 20 2021
Starting with offset 1 = binomial transform of A003945: (1, 3, 6, 12, 24, ...) and double bt of (1, 2, 1, 2, 1, 2, ...); equals polcoeff inverse of (1, -4, 3, 0, 0, 0, ...). - Gary W. Adamson, May 28 2009
Also the constant of the polynomials C(x) = 3x + 1 that form a sequence by performing this operation repeatedly and taking the result at each step as the input at the next. - Nishant Shukla (n.shukla722(AT)gmail.com), Jul 11 2009
It appears that this is A120444(3^n-1) = A004125(3^n) - A004125(3^n-1), where A004125 is the sum of remainders of n mod k for k = 1, 2, 3, ..., n. - John W. Layman, Jul 29 2009
Subsequence of A134025; A171960(a(n)) = a(n). - Reinhard Zumkeller, Jan 20 2010
Let A be the Hessenberg matrix of order n, defined by: A[1,j] = 1, A[i, i] := 3, (i > 1), A[i, i-1] = -1, and A[i, j] = 0 otherwise. Then, for n >= 1, a(n) = det(A). - Milan Janjic, Jan 27 2010
This is the sequence A(0, 1; 2, 3; 2) = A(0, 1; 4, -3; 0) of the family of sequences [a, b:c, d:k] considered by Gary Detlefs, and treated as A(a, b; c, d; k) in the Wolfdieter Lang link given below. - Wolfdieter Lang, Oct 18 2010
It appears that if s(n) is a first order rational sequence of the form s(0) = 0, s(n) = (2*s(n-1)+1)/(s(n-1)+2), n > 0, then s(n)= a(n)/(a(n)+1). - Gary Detlefs, Nov 16 2010
This sequence also describes the total number of moves to solve the [RED ; BLUE ; BLUE] or [RED ; RED ; BLUE] pre-colored Magnetic Towers of Hanoi puzzle (cf. A183111 - A183125).
From Adi Dani, Jun 08 2011: (Start)
a(n) is number of compositions of odd numbers into n parts less than 3. For example, a(3) = 13 and there are 13 compositions odd numbers into 3 parts < 3:
   1: (0, 0, 1), (0, 1, 0), (1, 0, 0);
   3: (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0), (1, 1, 1);
   5: (1, 2, 2), (2, 1, 2), (2, 2, 1).
(End)
Pisano period lengths:  1, 2, 1, 2, 4, 2, 6, 4, 1, 4, 5, 2, 3, 6, 4, 8, 16, 2, 18, 4, ... . - R. J. Mathar, Aug 10 2012
a(n) is the total number of holes (triangles removed) after the n-th step of a Sierpiński triangle production. - Ivan N. Ianakiev, Oct 29 2013
a(n) solves Sum_{j = a(n) + 1 .. a(n+1)} j = k^2 for some integer k, given a(0) = 0 and requiring smallest a(n+1) > a(n). Corresponding k = 3^n. - Richard R. Forberg, Mar 11 2015
a(n+1) equals the number of words of length n over {0, 1, 2, 3} avoiding 01, 02 and 03. - Milan Janjic, Dec 17 2015
For n >= 1, a(n) is also the total number of words of length n, over an alphabet of three letters, such that one of the letters appears an odd number of times (See A006516 for 4 letter words, and the Balakrishnan reference there). - Wolfdieter Lang, Jul 16 2017
Also, the number of maximal cliques, maximum cliques, and cliques of size 4 in the n-Apollonian network. - Andrew Howroyd, Sep 02 2017
For n > 1, the number of triangles (cliques of size 3) in the (n-1)-Apollonian network. - Andrew Howroyd, Sep 02 2017
a(n) is the largest number that can be represented with n trits in balanced ternary. Correspondingly, -a(n) is the smallest number that can be represented with n trits in balanced ternary. - Thomas König, Apr 26 2020
These form Sierpinski nesting-stars, which alternate pattern on 3^n+1/2 star numbers A003154, based on the square configurations of 9^n. The partial sums of 3^n are delineated according to the geometry of a hexagram, see illustrations in links. (3*a(n-1) + 1) create Sierpinski-anti-triangles, representing the number of holes in a (n+1) Sierpinski triangle (see illustrations). - John Elias, Oct 18 2021
For n > 1, a(n) is the number of iterations necessary to calculate the hyperbolic functions with CORDIC. - Mathias Zechmeister, Jul 26 2022
a(n) is the least number k such that A065363(k) = n. - Amiram Eldar, Sep 03 2022
For all n >= 0, Sum_{k=a(n)+1..a(n+1)} 1/k < Sum_{j=a(n+1)+1..a(n+2)} 1/j. These are the minimal points which partition the infinite harmonic series into a monotonically increasing sequence. Each partition approximates log(3) from below as n tends to infinity. - Joseph Wheat, Apr 15 2023
a(n) is also the number of 3-cycles in the n-Dorogovtsev-Goltsev-Mendes graph (using the convention the 0-Dorogovtsev-Goltsev-Mendes graph is P_2). - Eric W. Weisstein, Dec 06 2023

			There are 4 3-block bicoverings of a 3-set: {{1, 2, 3}, {1, 2}, {3}}, {{1, 2, 3}, {1, 3}, {2}}, {{1, 2, 3}, {1}, {2, 3}} and {{1, 2}, {1, 3}, {2, 3}}.
Ternary........Decimal
0.................0
1.................1
11................4
111..............13
1111.............40 etc. - _Zerinvary Lajos_, Jan 14 2007
There are altogether a(3) = 13 three letter words over {A,B,C} with say, A, appearing an odd number of times: AAA; ABC, ACB, ABB, ACC; BAC, CAB, BAB, CAC; BCA, CBA, BBA, CCA. - _Wolfdieter Lang_, Jul 16 2017

J. G. Mauldon, Strong solutions for the counterfeit coin problem, IBM Research Report RC 7476 (#31437) 9/15/78, IBM Thomas J. Watson Research Center, P. O. Box 218, Yorktown Heights, N. Y. 10598.
Paulo Ribenboim, The Book of Prime Number Records, Springer-Verlag, NY, 2nd ed., 1989, p. 60.
Paulo Ribenboim, The Little Book of Big Primes, Springer-Verlag, NY, 1991, p. 53.
Amir Sapir, The Tower of Hanoi with Forbidden Moves, The Computer J. 47 (1) (2004) 20, case three-in-a row, sequence a(n).
Robert Sedgewick, Algorithms, 1992, pp. 109.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Paolo Xausa, Table of n, a(n) for n = 0..2000 (terms 0..200 from T. D. Noe)
A. Abdurrahman, CM Method and Expansion of Numbers, arXiv:1909.10889 [math.NT], 2019.
Max A. Alekseyev and Toby Berger, Solving the Tower of Hanoi with Random Moves. In: J. Beineke, J. Rosenhouse (eds.) The Mathematics of Various Entertaining Subjects: Research in Recreational Math, Princeton University Press, 2016, pp. 65-79. ISBN 978-0-691-16403-8
Arcytech, The Sierpinski Triangle Fractal.
Andrei Asinowski and Michaela A. Polley, Patterns in rectangulations. Part I: T-like patterns, inversion sequence classes I(010, 101, 120, 201) and I(011, 201), and rushed Dyck paths, arXiv:2501.11781 [math.CO], 2025. See p. 26.
Jean-Luc Baril and Helmut Prodinger, Enumeration of partial Lukasiewicz paths, arXiv:2205.01383 [math.CO], 2022.
Beáta Bényi and Toshiki Matsusaka, Extensions of the combinatorics of poly-Bernoulli numbers, arXiv:2106.05585 [math.CO], 2021.
Göksal Bilgici and Tuncay Deniz Şentürk, Some Addition Formulas for Fibonacci, Pell and Jacobsthal Numbers, Annales Mathematicae Silesianae (2019) Vol. 33, 55-65.
Carlos M. da Fonseca and Anthony G. Shannon, A formal operator involving Fermatian numbers, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 491-498.
Sung-Hyuk Cha, On Complete and Size Balanced k-ary Tree Integer Sequences, International Journal of Applied Mathematics and Informatics, Issue 2, Volume 6, 2012, pp. 67-75. - From _N. J. A. Sloane_, Dec 24 2012
Shaoshi Chen, Hanqian Fang, Sergey Kitaev, and Candice X.T. Zhang, Patterns in Multi-dimensional Permutations, arXiv:2411.02897 [math.CO], 2024. See pp. 17, 26.
Nachum Dershowitz, Between Broadway and the Hudson: A Bijection of Corridor Paths, arXiv:2006.06516 [math.CO], 2020.
Roger B. Eggleton, Maximal Midpoint-Free Subsets of Integers, International Journal of Combinatorics Volume 2015, Article ID 216475, 14 pages.
John Elias, Sierpinski Nesting Stars, Stars of 3*9^n-1/2, Stars of 9^n-1/2, Sierpinski Anti-Triangles.
Graham Everest, Shaun Stevens, Duncan Tamsett and Tom Ward, Primes generated by recurrence sequences, Amer. Math. Monthly, Vol. 114, No. 5 (2007), pp. 417-431.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 372.
Takao Komatsu, Some recurrence relations of poly-Cauchy numbers, J. Nonlinear Sci. Appl., (2019) Vol. 12, Issue 12, 829-845.
G. Kreweras, Sur les éventails de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #15 (1970), 3-41. [Annotated scanned copy]
H. V. Krishna and N. J. A. Sloane, Correspondence, 1975.
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.
Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences.
László Németh, The trinomial transform triangle, J. Int. Seqs., Vol. 21 (2018), Article 18.7.3. Also arXiv:1807.07109 [math.NT], 2018.
László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
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.
Yash Puri and Thomas Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
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.
A. G. Shannon, Letter to N. J. A. Sloane, Dec 06 1974.
Morgan Ward, Note on divisibility sequences, Bull. Amer. Math. Soc., 42 (1936), 843-845.
Eric Weisstein's World of Mathematics, Apollonian Network.
Eric Weisstein's World of Mathematics, Dorogovtsev-Goltsev-Mendes Graph.
Eric Weisstein's World of Mathematics, Graph Cycle.
Eric Weisstein's World of Mathematics, Maximal Clique.
Eric Weisstein's World of Mathematics, Maximum Clique.
Eric Weisstein's World of Mathematics, Mephisto Waltz Sequence.
Eric Weisstein's World of Mathematics, Repunit.
Eric Weisstein's World of Mathematics, Weighing.
Thomas Wieder, The number of certain k-combinations of an n-set, Applied Mathematics Electronic Notes, Vol. 8 (2008).
K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. Math., Vol. 3 (1892), 265-284.
Index entries for sequences related to sorting
Index entries for linear recurrences with constant coefficients, signature (4,-3)

Sequences used for Shell sort: A033622, A003462, A036562, A036564, A036569, A055875.
Cf. A179526 (repeats), A113047 (characteristic function).
Cf. A000225, A000392, A004125, A014753, A028491 (indices of primes), A059443 (column k = 3), A065363, A097933, A120444, A321872 (sum reciprocals).
Cf. A064099 (minimal number of weightings to detect lighter or heavier coin among n coins).
Cf. A039755 (column k = 1).
Cf. A006516 (binomial transform, and special 4 letter words).
Cf. A341590.
Cf. A003462(n) (3-cycles), A367967(n) (5-cycles), A367968(n) (6-cycles).

A003462:=List([0..30],n->(3^n-1)/2); # Muniru A Asiru, Sep 27 2017

a003462 = (`div` 2) . (subtract 1) . (3 ^)
a003462_list = iterate ((+ 1) . (* 3)) 0  -- Reinhard Zumkeller, May 09 2012

[(3^n-1)/2: n in [0..30]]; // Vincenzo Librandi, Feb 21 2015

A003462 := n-> (3^n - 1)/2: seq(A003462(n), n=0..30);
A003462:=1/(3*z-1)/(z-1); # Simon Plouffe in his 1992 dissertation

(3^Range[0, 30] - 1)/2 (* Harvey P. Dale, Jul 13 2011 *)
LinearRecurrence[{4, -3}, {0, 1}, 30] (* Harvey P. Dale, Jul 13 2011 *)
Accumulate[3^Range[0, 30]] (* Alonso del Arte, Sep 10 2017 *)
CoefficientList[Series[x/(1 - 4x + 3x^2), {x, 0, 30}], x] (* Eric W. Weisstein, Sep 28 2017 *)
Table[FromDigits[PadRight[{},n,1],3],{n,0,30}] (* Harvey P. Dale, Jun 01 2022 *)

A003462(n):=(3^n - 1)/2$
makelist(A003462(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI
```
a(n)=(3^n-1)/2
```

concat(0, Vec(x/((1-x)*(1-3*x)) + O(x^30))) \\ Altug Alkan, Nov 01 2015

[(3^n - 1)/2 for n in range(0,30)] # Zerinvary Lajos, Jun 05 2009

G.f.: x/((1-x)*(1-3*x)).
a(n) = 4*a(n-1) - 3*a(n-2), n > 1. a(0) = 0, a(1) = 1.
a(n) = 3*a(n-1) + 1, a(0) = 0.
E.g.f.: (exp(3*x) - exp(x))/2. - Paul Barry, Apr 11 2003
a(n+1) = Sum_{k = 0..n} binomial(n+1, k+1)*2^k. - Paul Barry, Aug 20 2004
a(n) = Sum_{i = 0..n-1} 3^i, for n > 0; a(0) = 0.
a(n) = A125118(n, 2) for n > 1. - Reinhard Zumkeller, Nov 21 2006
a(n) = StirlingS2(n+1, 3) + StirlingS2(n+1, 2). - Ross La Haye, Jan 10 2008
a(n) = Sum_{k = 0..n} A106566(n, k)*A106233(k). - Philippe Deléham, Oct 30 2008
a(n) = 2*a(n-1) + 3*a(n-2) + 2, n > 1. - Gary Detlefs, Jun 21 2010
a(n) = 3*a(n-1) + a(n-2) - 3*a(n-3) = 5*a(n-1) - 7*a(n-2) + 3*a(n-3), a(0) = 0, a(1) = 1, a(2) = 4. Observation by G. Detlefs. See the W. Lang comment and link. - Wolfdieter Lang, Oct 18 2010
A008344(a(n)) = 0, for n > 1. - Reinhard Zumkeller, May 09 2012
A085059(a(n)) = 1 for n > 0. - Reinhard Zumkeller, Jan 31 2013
G.f.: Q(0)/2 where Q(k) = 1 - 1/(9^k - 3*x*81^k/(3*x*9^k - 1/(1 - 1/(3*9^k - 27*x*81^k/(9*x*9^k - 1/Q(k+1)))))); (continued fraction ). - Sergei N. Gladkovskii, Apr 12 2013
a(n) = A001065(3^n) where A001065(m) is the sum of the proper divisors of m for positive integer m. - Chayim Lowen, Mar 03 2015
a(n) = A000244(n) - A007051(n) = A007051(n)-1. - Yuchun Ji, Oct 23 2018
Sum_{n>=1} 1/a(n) = A321872. - Amiram Eldar, Nov 18 2020

More terms from Michael Somos
Corrected my comment of Jan 10 2008. - Ross La Haye, Oct 29 2008
Removed comment that duplicated a formula. - Joerg Arndt, Mar 11 2010

A183125 Magnetic Tower of Hanoi, total number of moves, generated by a certain algorithm, yielding a "forward moving" non-optimal solution of the [NEUTRAL ; NEUTRAL ; NEUTRAL] pre-colored puzzle.

Original entry on oeis.org

0, 1, 4, 11, 30, 83, 236, 687, 2026, 6027, 18008, 53927, 161654, 484803, 1454212, 4362399, 13086914, 39260411, 117780848, 353342103, 1060025806, 3180076851, 9540229916, 28620689039, 85862066330, 257586198123, 772758593416, 2318275779207, 6954827336486, 20864482008227, 62593446023348, 187780338068607, 563341014204274, 1690023042611163

Offset: 0

Uri Levy, Jan 08 2011

The Magnetic Tower of Hanoi puzzle is described in link 1 listed below. The Magnetic Tower is pre-colored. Pre-coloring is [NEUTRAL ; NEUTRAL ; NEUTRAL], given in [Source ; Intermediate ; Destination] order. The solution algorithm producing the presented sequence is NOT optimal. The particular "61" algorithm solving the puzzle at hand is not explicitly presented in any of the referenced papers. For the optimal solution of the Magnetic Tower of Hanoi puzzle with the given pre-coloring configuration (the "natural" or "free" Magnetic Tower) see A183117 and A183118. Optimal solutions are discussed and their optimality is proved in link 2 listed below.

Large N limit of the sequence is 0.5*(197/324)*3^N ~ 0.5*0.61*3^N. Series designation: S61(n).

Uri Levy, The Magnetic Tower of Hanoi, Journal of Recreational Mathematics, Volume 35 Number 3 (2006), 2010, pp 173.

Muniru A Asiru, Table of n, a(n) for n = 0..2020
Uri Levy, The Magnetic Tower of Hanoi, arxiv:1003.0225 [math.CO], 2010.
Uri Levy, Magnetic Towers of Hanoi and their Optimal Solutions, arXiv:1011.3843 [math.CO], 2010.
Uri Levy, to play The Magnetic Tower of Hanoi, web applet [Broken link]
Index entries for linear recurrences with constant coefficients, signature (5,-6,-2,7,-3).

A183123 is an integer sequence generated by another non-optimal algorithm solving the "free" [NEUTRAL ; NEUTRAL ; NEUTRAL] Magnetic Tower of Hanoi puzzle.
A003462 "Partial sums of A000244" is the sequence (also) describing the total number of moves solving [RED ; BLUE ; BLUE] or [RED ; RED ; BLUE] pre-colored Magnetic Tower of Hanoi puzzle.
Cf. A183111 - A183125.

a:=[30, 83, 236, 687, 2026];; for n in [6..30] do a[n]:=5*a[n-1]-6*a[n-2] -2*a[n-3]+7*a[n-4]-3*a[n-5]; od; Concatenation([0, 1, 4, 11], a); # G. C. Greubel, Dec 04 2018

I:=[0,1,4,11,30,83,236,687,2026]; [n le 9 select I[n] else 5*Self(n-1)-6*Self(n-2)-2*Self(n-3)+7*Self(n-4)-3*Self(n-5): n in [1..35]]; // Vincenzo Librandi, Dec 04 2018

m:=30; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!( (-4*x^8 -2*x^6 +x^4 -3*x^3 -x^2 +x)/(3*x^5 -7*x^4 +2*x^3 +6*x^2 -5*x +1))); // G. C. Greubel, Dec 04 2018

seq(coeff(series((-4*x^8-2*x^6+x^4-3*x^3-x^2+x)/(3*x^5-7*x^4+2*x^3+6*x^2-5*x+1),x,n+1), x, n), n = 0 .. 35); # Muniru A Asiru, Dec 04 2018

Join[{0, 1, 4, 11}, LinearRecurrence[{5, -6, -2, 7, -3}, {30, 83, 236, 687, 2026}, 30]] (* Jean-François Alcover, Dec 04 2018 *)
CoefficientList[Series[(- 4 x^8 - 2 x^6 + x^4 - 3 x^3 - x^2 + x) / (3 x^5 - 7 x^4 + 2 x^3 + 6 x^2 - 5 x + 1), {x, 0, 33}], x] (* Vincenzo Librandi, Dec 04 2018 *)

my(x='x+O('x^30)); concat([0], Vec((-4*x^8 -2*x^6 +x^4 -3*x^3 -x^2 +x)/(3*x^5 -7*x^4 +2*x^3 +6*x^2 -5*x +1))) \\ G. C. Greubel, Dec 04 2018

s=((-4*x^8 -2*x^6 +x^4 -3*x^3 -x^2 +x)/(3*x^5 -7*x^4 +2*x^3 +6*x^2 -5*x +1)).series(x, 30); s.coefficients(x, sparse=False) # G. C. Greubel, Dec 04 2018

G.f.: (-4*x^8 -2*x^6 +x^4 -3*x^3 -x^2 +x)/(3*x^5 -7*x^4 +2*x^3 +6*x^2 -5*x +1).
a(n) = +5*a(n-1)-6*a(n-2)-2*a(n-3)+7*a(n-4)-3*a(n-5).
(a(n) = S61(n) as in referenced paper):
a(n) = 3*a(n-1) - 2*n^2 + 17*n - 43 ; n even ; n >= 6.
a(n) = 3*a(n-1) - 2*n^2 + 17*n - 42 ; n odd ; n >= 5.
a(n) = S64(n-1) + S64(n-2) + S75(n-3) + 4*3^(n-3) + 2 ; n >= 3.
S64(n) and S75(n) refer to the integer sequences described by A183121 and A183119 respectively.
a(n) = 0.5*(197/324)*3^n + n^2 - 5.5*n + 91/8; n even; n >= 4.
a(n) = 0.5*(197/324)*3^n + n^2 - 5.5*n + 93/8; n odd; n >= 5.

More terms from Jean-François Alcover, Dec 04 2018

A122983 a(n) = (2 + (-1)^n + 3^n)/4.

Original entry on oeis.org

1, 1, 3, 7, 21, 61, 183, 547, 1641, 4921, 14763, 44287, 132861, 398581, 1195743, 3587227, 10761681, 32285041, 96855123, 290565367, 871696101, 2615088301, 7845264903, 23535794707, 70607384121, 211822152361, 635466457083

Offset: 0

Paul Barry, Sep 22 2006

Old definition was: "Binomial transform of aeration of A081294".
Binomial transform is A063376.
A122983 = (1,1,3,7,1,1,3,7,...) mod 10. - M. F. Hasler, Feb 25 2008
Equals row sums of triangle A158301. - Gary W. Adamson, Mar 15 2009
a(n) = the number of ternary sequences of length n where the numbers of (0's, 1's) are both even. A015518 covers the (odd, even) and (even, odd) cases, and A081251 covers (odd, odd). - Toby Gottfried, Apr 18 2010
This sequence also describes the number of moves of the k-th disk solving (non-optimally) the [RED ; NEUTRAL ; BLUE] pre-colored Magnetic Tower of Hanoi (MToH) puzzle. The sequence A183119 is the partial sums of the sequence in question (obviously describing the total number of moves associated with the specific solution algorithm). For other MToH-related sequences, Cf. A183111 - A183125.
Let B=[1,sqrt(2),0; sqrt(2),1,sqrt(2); 0,sqrt(2),1] be a 3 X 3 matrix. Then a(n)=[B^n](1,1), n=0,1,2,.... - _L. Edson Jeffery, Dec 21 2011
Also the domination number of the n-Hanoi graph. - Eric W. Weisstein, Jun 16 2017
Also the matching number of the n-Sierpinski gasket graph. - Eric W. Weisstein, Jun 17 2017
Let M = [1,1,1,0; 1,1,0,1; 1,0,1,1; 0,1,1,1], a 4 X 4 matrix. Then a(n) is the upper left entry in M^n. - Philippe Deléham, Aug 23 2020
Also the lower matching number (=independent domination number) of the n-Hanoi graph. - Eric W. Weisstein, Aug 01 2023

M. F. Hasler, Table of n, a(n) for n = 0..199.
Ji Young Choi, A Generalization of Collatz Functions and Jacobsthal Numbers, J. Int. Seq., Vol. 21 (2018), Article 18.5.4.
Alexander Diaz-Lopez, Pamela E. Harris, Erik Insko, and Darleen Perez-Lavin, Peaks Sets of Classical Coxeter Groups, arXiv preprint, arXiv:1505.04479 [math.GR], 2015.
A. M. Hinz, S. Klavžar, U. Milutinović, and C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 99. Book's website
Uri Levy, The Magnetic Tower of Hanoi, arXiv:1003.0225 [math.CO], 2010.
Eric Weisstein's World of Mathematics, Domination Number.
Eric Weisstein's World of Mathematics, Hanoi Graph.
Eric Weisstein's World of Mathematics, Lower Independence Number.
Eric Weisstein's World of Mathematics, Matching Number.
Eric Weisstein's World of Mathematics, Sierpiński Gasket Graph.
Index entries for linear recurrences with constant coefficients, signature (3,1,-3).

Cf. a(j+1) = A137822(2^j) and these are the record values of A137822.

Cf. A054879 (bisection), A066443 (bisection). Row sums of A158303.

A122983 := n -> ceil(3^n/4); 'A122983(n)' $ n=0..22; # M. F. Hasler, Feb 25 2008
a[ -1]:=1:a[0]:=1:a[1]:=3:for n from 2 to 50 do a[n]:=2*a[n-1]+3*a[n-2]-2 od: seq(a[n], n=-1..25); # Zerinvary Lajos, Apr 28 2008

CoefficientList[Series[(1 - 2 x - x^2)/((1 - x) (1 + x) (1 - 3 x)), {x, 0, 40}], x] (* Harvey P. Dale, Sep 03 2013 *)
LinearRecurrence[{3, 1, -3}, {1, 1, 3}, 40] (* Harvey P. Dale, Sep 03 2013 *)
Table[(2 + (-1)^n + 3^n)/4, {n, 0, 20}] (* Eric W. Weisstein, Jun 16 2017 *)
Table[Floor[3^n/4] + 1, {n, 0, 20}] (* Eric W. Weisstein, Jan 17 2018 *)
Floor[3^Range[0, 20]/4] + 1 (* Eric W. Weisstein, Jan 17 2018 *)

A122983(n)=3^n\4+1 \\ M. F. Hasler, Feb 25 2008

def A122983(n): return (1 if n&1 else 3)+3**n>>2 # Chai Wah Wu, Apr 12 2023

From Paul Barry, Jun 14 2007: (Start)
G.f.: (1-2*x-x^2)/((1-x)*(1+x)*(1-3*x));
a(n) = 3^n/4+(-1)^n/4+1/2;
E.g.f.: cosh(x)^2*exp(x). (End)
a(n) = 3*a(n-1) + a(n-2) - 3*a(n-3); a(0)=1, a(1)=1, a(2)=3. - Harvey P. Dale, Sep 03 2013
E.g.f.: Q(0)/2, where Q(k) = 1 + 3^k/( 2 - 2*(-1)^k/( 3^k + (-1)^k - 2*x*3^k/( 2*x + (k+1)*(-1)^k/Q(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Dec 22 2013
a(2*n) = 3*a(2*n-1); a(2*n+1) = 3*a(2*n) - 2. - Philippe Deléham, Aug 23 2020

Extended and corrected (existing Maple code) by M. F. Hasler, Feb 25 2008

Description changed to formula by Eric W. Weisstein, Jun 16 2017

A104743 a(n) = 3^n + n.

Original entry on oeis.org

1, 4, 11, 30, 85, 248, 735, 2194, 6569, 19692, 59059, 177158, 531453, 1594336, 4782983, 14348922, 43046737, 129140180, 387420507, 1162261486, 3486784421, 10460353224, 31381059631, 94143178850, 282429536505, 847288609468

Offset: 0

Zak Seidov, Mar 23 2005

This sequence also describes the total number of moves solving (non-optimally) the [RED ; NEUTRAL ; NEUTRAL] or [NEUTRAL ; NEUTRAL ; BLUE] pre-colored Magnetic Tower of Hanoi puzzle (see the "CROSSREFS" in A183121). For other Magnetic Tower of Hanoi related sequences, cf. A183111 - A183125.

Form an infinite array with m(0,k) = 2*k+1 and m(n,k) = Sum_{r=0..n-1} m(r,k) + Sum_{c=0..k-1} m(n,c), being the sum of the terms above m(n,k) plus those terms to the left of m(n,k). The first row is A005408. The second row is A033484. The first column is A011782. The sum of the terms in the n-th antidiagonal is a(n). - J. M. Bergot, Jun 07 2013

G. C. Greubel, Table of n, a(n) for n = 0..1000
Uri Levy, The Magnetic Tower of Hanoi, arXiv:1003.0225 [math.CO], 2010.
Index entries for linear recurrences with constant coefficients, signature (5,-7,3).

Cf. A103537.

List([0..40], n-> 3^n +n ); # G. C. Greubel, May 21 2019

[3^n +n: n in [0..40]]; // G. C. Greubel, May 21 2019

Table[3^n +n, {n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Jan 18 2009, modified by G. C. Greubel, May 21 2019 *)
LinearRecurrence[{5,-7,3},{1,4,11},30] (* Harvey P. Dale, Aug 01 2020 *)

{a(n) = 3^n + n}; \\ G. C. Greubel, May 21 2019

[3^n +n for n in (0..40)] # G. C. Greubel, May 21 2019

a(n) = n + 3^n.
From Colin Barker, Jan 25 2012: (Start)
a(n) = 5*a(n-1) - 7*a(n-2) + 3*a(n-3).
G.f.: (1+x)*(1-2*x) / ((1-3*x)*(1-x)^2). (End)
E.g.f.: x*exp(x) + exp(3*x). - G. C. Greubel, May 21 2019

More terms from Vladimir Joseph Stephan Orlovsky, Jan 18 2009

A100702 Number of layers of dough separated by butter in successive foldings of croissant dough.

Original entry on oeis.org

1, 3, 7, 19, 55, 163, 487, 1459, 4375, 13123, 39367, 118099, 354295, 1062883, 3188647, 9565939, 28697815, 86093443, 258280327, 774840979, 2324522935, 6973568803, 20920706407, 62762119219, 188286357655, 564859072963

Offset: 0

Daniel Wolf (djwolf1(AT)axelero.hu), Dec 09 2004

At each trebling of layers following the first, two sets of layers, not separated from their neighbors by butter, are combined. Traditional patisserie stops at 55 layers, but forgetful chefs have been known to make additional folds to 163 layers.
This sequence also describes the number of moves of the k-th disk solving (non-optimally) the [RED ; NEUTRAL ; NEUTRAL] or [NEUTRAL ; NEUTRAL ; BLUE] pre-colored Magnetic Tower of Hanoi puzzle (see the "CROSSREFS" in A183120). For other Magnetic Tower of Hanoi related sequences cf. A183111-A183125.
Same as A052919 except first term is 1, not 2. - Omar E. Pol, Feb 20 2011

J. Child and M. Beck, Mastering the Art of French Cooking, Vol. 2

Uri Levy, The Magnetic Tower of Hanoi, arXiv:1003.0225 [math.CO], 2010.
Index entries for linear recurrences with constant coefficients, signature (4, -3).

Cf. A052919.

Join[{1}, LinearRecurrence[{4, -3}, {3, 7}, 25]] (* Jean-François Alcover, Jul 28 2018 *)

a(n)=([0,1; -3,4]^n*[1;3])[1,1] \\ Charles R Greathouse IV, Jan 28 2018

For n > 1, a(n) = 3*a(n-1) - 2.
From R. J. Mathar, Jun 30 2009: (Start)
a(n) = 1 + 2*3^(n-1), n > 0.
a(n) = 4*a(n-1) - 3*a(n-2), n > 2.
G.f.: -(1+x)*(2*x-1)/((3*x-1)*(x-1)). (End)

A183119 Magnetic Tower of Hanoi, total number of moves generated by a certain algorithm, yielding a "forward moving" non-optimal solution of the [RED ; NEUTRAL ; BLUE] pre-colored puzzle.

Original entry on oeis.org

0, 1, 4, 11, 32, 93, 276, 823, 2464, 7385, 22148, 66435, 199296, 597877, 1793620, 5380847, 16142528, 48427569, 145282692, 435848059, 1307544160, 3922632461, 11767897364, 35303692071, 105911076192, 317733228553, 953199685636, 2859599056883, 8578797170624, 25736391511845

Offset: 0

Uri Levy, Jan 03 2011

The Magnetic Tower of Hanoi puzzle is described in link 1 listed below. The Magnetic Tower is pre-colored. Pre-coloring is [RED ; NEUTRAL ; BLUE], given in [Source ; Intermediate ; Destination] order. The solution algorithm producing the presented sequence is NOT optimal. The particular "75" algorithm solving the puzzle at hand is presented in a paper referenced by link 1 listed below. For the optimal solution of the Magnetic Tower of Hanoi puzzle with the given pre-coloring configuration see A183113 and A183114. Optimal solutions are discussed and their optimality is proved in link 2 listed below.

Large N limit of the sequence is 0.5*(3/4)*3^N = 0.5*0.75*3^N. Series designation: S75(n).

Uri Levy, The Magnetic Tower of Hanoi, Journal of Recreational Mathematics, Volume 35 Number 3 (2006), 2010, pp 173.

Muniru A Asiru, Table of n, a(n) for n = 0..2070
Uri Levy, The Magnetic Tower of Hanoi, arXiv:1003.0225 [math.CO], 2010.
Uri Levy, Magnetic Towers of Hanoi and their Optimal Solutions, arXiv:1011.3843 [math.CO], 2010.
Uri Levy, to play The Magnetic Tower of Hanoi, web applet [Broken link]
Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,3).

A122983 - "Binomial transform of aeration of A081294" is an "original" sequence (also) describing the number of moves of disk number k, solving the pre-colored puzzle at hand when executing the "75" algorithm mentioned above and presented in the paper referenced by link 1 above. The integer sequence listed above is the partial sums of the A122983 original sequence.
A003462 "Partial sums of A000244" is the sequence (also) describing the total number of moves solving [RED ; BLUE ; BLUE] or [RED ; RED ; BLUE] pre-colored Magnetic Tower of Hanoi puzzle.
Cf. A183111-A183125.

[3^(n+1)/8+(n-1)/2+(-1)^n/8: n in [0..30]]; // Vincenzo Librandi, Dec 04 2018

seq(coeff(series(x*(3*x^2-1)/((1+x)*(3*x-1)*(x-1)^2),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Dec 04 2018

LinearRecurrence[{4, -2, -4, 3}, {0, 1, 4, 11}, 30] (* Jean-François Alcover, Dec 04 2018 *)
Table[3^(n + 1) / 8 + (n - 1) / 2 + (-1)^n / 8, {n, 0, 30}] (* Vincenzo Librandi, Dec 04 2018 *)

a(n) = 3^(n+1)/8 + (n-1)/2 + (-1)^n/8 \\ Charles R Greathouse IV, Jun 11 2015

G.f.: x*(-1+3*x^2) / ( (1+x)*(3*x-1)*(x-1)^2 ).
(a(n)=S75(n) in referenced paper):
a(n) = 3*a(n-1) - n + 3; n even; n >= 2
a(n) = 3*a(n-1) - n + 2; n odd; n >= 1
a(n) = a(n-2) + 3^(n-1) + 1; n >= 2
a(n) = 3^(n+1)/8 + (n-1)/2 +(-1)^n/8.

More terms from Jean-François Alcover, Dec 04 2018

A183117 Magnetic Tower of Hanoi, number of moves of disk number k, optimally solving the [NEUTRAL ; NEUTRAL ; NEUTRAL] pre-colored puzzle.

Original entry on oeis.org

0, 1, 3, 7, 19, 53, 153, 451, 1339, 3997, 11961, 35835, 107435, 322197, 966425, 2899027, 8696699, 26089517, 78267673, 234801675, 704402987, 2113205861, 6339612857, 19018831395, 57056483259, 171169433149, 513508274169, 1540524784027, 4621574293547, 13864722791605, 41594168239321

Offset: 0

Uri Levy, Dec 31 2010

nonn

The Magnetic Tower of Hanoi puzzle is described in link 1 listed below. The Magnetic Tower is pre-colored. Pre-coloring is [NEUTRAL ; NEUTRAL ; NEUTRAL],  given in [Source ; Intermediate ; Destination] order. Thus, the tower in this case is "natural" or "free". The solution algorithm producing the sequence is optimal (the sequence presented gives the minimum number of moves to solve the puzzle for the given pre-coloring configuration). Optimal solutions are discussed and their optimality is proved in link 2 listed below.
Disk numbering is from largest disk (k = 1) to smallest disk (k = N).
The above-listed "original" sequence generates a "partial-sums" sequence - describing the total number of moves required to solve the puzzle.
Number of moves of disk k, for large k, is close to (20/33)*3^(k-1) ~ 0.606*3^(k-1). Series designation: P606(k).

Uri Levy, The Magnetic Tower of Hanoi, Journal of Recreational Mathematics, Volume 35 Number 3 (2006), 2010, pp. 173.

Uri Levy, The Magnetic Tower of Hanoi, arXiv:1003.0225 [math.CO], 2010.
Uri Levy, Magnetic Towers of Hanoi and their Optimal Solutions, arXiv:1011.3843 [math.CO], 2010.
Web applet to play The Magnetic Tower of Hanoi [Broken link]

A000244 "Powers of 3" is the sequence (also) describing the number of moves of the k-th disk solving [RED ; BLUE ; BLUE] or [RED ; RED ; BLUE] pre-colored Magnetic Tower of Hanoi puzzle.

A183111 through A183125 are related sequences, all associated with various solutions of the pre-coloring variations of the Magnetic Tower of Hanoi.

L1 = Root[-2 - # + #^3&, 1];
L2 = Root[-2 - # + #^3&, 3];
L3 = Root[-2 - # + #^3&, 2];
AP = Root[-2 - 9# - 52 #^2 + 572 #^3&, 1];
BP = Root[-2 - 9# - 52 #^2 + 572 #^3&, 3];
CP = Root[-2 - 9# - 52 #^2 + 572 #^3&, 2];
a[0] = 0;
a[n_] := (1/2) AP (L1+1)^2 L1^(n-1) + (1/2) BP (L2+1)^2 L2^(n-1) + (1/2) CP (L3+1)^2 L3^(n-1) + (20 3^(n-1))/33;
Table[a[n] // Round, {n, 0, 30}] (* Jean-François Alcover, Dec 03 2018 *)

G.f. appears to be -x*(1+x)*(2*x^3+2*x^2+x-1)/((3*x-1)*(2*x^3+x^2-1)) with a(n) = 3*a(n-1) + a(n-2) - a(n-3) - 6*a(n-4) for n > 5. - Joerg Arndt, Jan 03 2011
Recurrence Relations (a(n)=P606(n) as in referenced paper):
P606(n) = P636(n-1) + P636(n-2) + P909(n-2) + 2*3^(n-3) ; n >= 3.
Note:  P636(n) and P909(n) refer to the integer sequences described by A183115 and A183111 respectively.
Closed-Form Expression:
Define:
λ1 = (1+sqrt(26/27))^(1/3) + (1-sqrt(26/27))^(1/3)
λ2 = -0.5*λ1 + 0.5*i*((sqrt(27) + sqrt(26))^(1/3) - (sqrt(27) - sqrt(26))^(1/3))
λ3 = -0.5*λ1 - 0.5*i*((sqrt(27) + sqrt(26))^(1/3) - (sqrt(27) - sqrt(26))^(1/3))
AP = ((1/11)*λ2*λ3 - (3/11)*(λ2 + λ3) + (9/11))/((λ2 - λ1)*(λ3 - λ1))
BP = ((1/11)*λ1*λ3 - (3/11)*(λ1 + λ3) + (9/11))/((λ1 - λ2)*(λ3 - λ2))
CP = ((1/11)*λ1*λ2 - (3/11)*(λ1 + λ2) + (9/11))/((λ2 - λ3)*(λ1 - λ3))
For n > 1:
P606(n) = (20/33)*3^(n-1) + 0.5*AP*((λ1+1)^2)*λ1^(n-1) + 0.5*BP*((λ2+1)^2)*λ2^(n-1) + 0.5*CP*(λ3+1)^2)*λ3^(n-1).

A183115 Magnetic Tower of Hanoi, number of moves of disk number k, optimally solving the [RED ; NEUTRAL ; NEUTRAL] or [NEUTRAL ; NEUTRAL ; BLUE] pre-colored puzzle.

Original entry on oeis.org

0, 1, 3, 7, 19, 55, 159, 471, 1403, 4191, 12551, 37615, 112787, 338279, 1014703, 3043911, 9131435, 27393839, 82180823, 246541407, 739622595, 2218865335, 6656592255, 19969771063, 59909304539, 179727900415, 539183681191, 1617551013071, 4852652992755, 14557958907655, 43673876615503

Offset: 0

Uri Levy, Dec 31 2010

nonn

The Magnetic Tower of Hanoi puzzle is described in link 1 listed below. The Magnetic Tower is pre-colored. Pre-coloring is [RED ; NEUTRAL ; NEUTRAL] or [NEUTRAL ; NEUTRAL ; BLUE],  given in [Source ; Intermediate ; Destination] order. The solution algorithm producing the sequence is optimal (the sequence presented gives the minimum number of moves to solve the puzzle for the given pre-coloring configurations). Optimal solutions are discussed and their optimality is proved in link 2 listed below.
Disk numbering is from largest disk (k = 1) to smallest disk (k = N)
The above-listed "original" sequence generates a "partial-sums" sequence - describing the total number of moves required to solve the puzzle.
Number of moves of disk k, for large k,  is close to (7/11)*3^(k-1) ~ 0.636*3^(k-1). Series designation: P636(k).

Uri Levy, "The Magnetic Tower of Hanoi", Journal of Recreational Mathematics, Volume 35 Number 3 (2006), 2010, pp. 173.

Uri Levy, The Magnetic Tower of Hanoi, arXiv:1003.0225 [math.CO], 2010.
Uri Levy, Magnetic Towers of Hanoi and their Optimal Solutions, arXiv:1011.3843 [math.CO], 2010.
Web applet to play The Magnetic Tower of Hanoi [Broken link]

A000244 "Powers of 3" is the sequence (also) describing the number of moves of the k-th disk solving [RED ; BLUE ; BLUE] or [RED ; RED ; BLUE] pre-colored Magnetic Tower of Hanoi puzzle.

A183111 through A183125 are related sequences, all associated with various solutions of the pre-coloring variations of the Magnetic Tower of Hanoi.

L1 = Root[-2 - # + #^3&, 1];
L2 = Root[-2 - # + #^3&, 3];
L3 = Root[-2 - # + #^3&, 2];
AP = Root[-2 - 9 # - 52 #^2 + 572 #^3&, 1];
BP = Root[-2 - 9 # - 52 #^2 + 572 #^3&, 3];
CP = Root[-2 - 9 # - 52 #^2 + 572 #^3&, 2];
a[0] = 0;
a[n_] := (7/11) 3^(n-1) + AP (L1+1) L1^(n-1) + BP (L2+1) L2^(n-1) + CP (L3+1) L3^(n-1);
Table[a[n] // Round, {n, 0, 30}] (* Jean-François Alcover, Dec 03 2018 *)

Recurrence Relations (a(n)=P636(n) as in referenced paper):
P636(n) = P636(n-1) + 2*P909(n-2) + 2*3^(n-3) ; n >= 3
Note:  P909(n-2) refers to the integer sequence described by A183111.
Closed-Form Expression:
Define:
λ1 = [1+sqrt(26/27)]^(1/3) +  [1-sqrt(26/27)]^(1/3)
λ2 = -0.5* λ1 + 0.5*i*{[sqrt(27)+sqrt(26)]^(1/3)- [sqrt(27)-sqrt(26)]^(1/3)}
λ3 = -0.5* λ1 - 0.5*i*{[sqrt(27)+sqrt(26)]^(1/3)- [sqrt(27)-sqrt(26)]^(1/3)}
AP = [(1/11)* λ2* λ3 - (3/11)*(λ2 + λ3) + (9/11)]/[( λ2 - λ1)*( λ3 - λ1)]
BP = [(1/11)* λ1* λ3 - (3/11)*(λ1 + λ3) + (9/11)]/[( λ1 - λ2)*( λ3 - λ2)]
CP = [(1/11)* λ1* λ2 - (3/11)*(λ1 + λ2) + (9/11)]/[( λ2 - λ3)*( λ1 - λ3)]
For n > 0: P636(n) = (7/11)*3^(n-1) + AP*(λ1+1)*λ1^(n-1) + BP*( λ2+1)*λ2^(n-1) + CP*(λ3+1)* λ3^(n-1)
G.f.: x*(1-3*x^2-4*x^3)/((1-3*x)*(1-x^2-2*x^3)). - Colin Barker, Jan 12 2012

More terms from Jean-François Alcover, Dec 03 2018

A183118 Magnetic Tower of Hanoi, total number of moves, optimally solving the [NEUTRAL ; NEUTRAL ; NEUTRAL] pre-colored puzzle.

Original entry on oeis.org

0, 1, 4, 11, 30, 83, 236, 687, 2026, 6023, 17984, 53819, 161254, 483451, 1449876, 4348903, 13045602, 39135119, 117402792, 352204467, 1056607454

Offset: 0

Uri Levy, Jan 01 2011

nonn

A.   The Magnetic Tower of Hanoi puzzle is described in link 1 listed below. The Magnetic Tower is pre-colored. Pre-coloring is [NEUTRAL ; NEUTRAL ; NEUTRAL], given in [Source ; Intermediate ; Destination] order. Thus, the tower in this case is "natural" or "free". The solution algorithm producing the sequence is optimal (the sequence presented gives the minimum number of moves to solve the "free" Magnetic Tower of Hanoi puzzle). Optimal solutions are discussed and their optimality is proved in link 2 listed below.
B.   Number of moves to solve the given puzzle, for large N, is close to 0.5*(20/33)*3^N ~ 0.5*0.606*3^(N). Series designation: S606(N).
C.   The large N ratio of number of moves to solve the [NEUTRAL ; NEUTRAL ; NEUTRAL] puzzle to the number of moves to solve the [RED ; BLUE ; BLUE] or [RED ; RED ; BLUE] puzzle is 20/33 or about 60.6% (see link 2).

"The Magnetic Tower of Hanoi", Uri Levy, Journal of Recreational Mathematics, Volume 35 Number 3 (2006), 2010, pp 173.

Uri Levy, The Magnetic Tower of Hanoi, arXiv:1003.0225 [math.CO], 2010.
Uri Levy, Magnetic Towers of Hanoi and their Optimal Solutions, arXiv:1011.3843 [math.CO], 2010.
Web applet, to play The Magnetic Tower of Hanoi
Index entries for linear recurrences with constant coefficients, signature (4,-2,-2,-5,6).

A183117 is an "original" sequence describing the number of moves of disk number k, optimally solving the pre-colored puzzle at hand. The integer sequence listed above is the partial sums of the A183117 original sequence.
A003462 "Partial sums of A000244" is the sequence (also) describing the total number of moves solving [RED ; BLUE ; BLUE] or [RED ; RED ; BLUE] pre-colored Magnetic Tower of Hanoi puzzle.
A183111 through A183125 are related sequences, all associated with various solutions of the pre-coloring variations of the Magnetic Tower of Hanoi.

Join[{0}, LinearRecurrence[{4, -2, -2, -5, 6}, {1, 4, 11, 30, 83}, 20]] (* Jean-François Alcover, Jan 28 2019 *)

G.f. x*(-2*x^4-4*x^3-3*x^2+1)/(-6*x^5+5*x^4+2*x^3+2*x^2-4*x+1).
Recurrence Relations (a(n)=S606(n) as in referenced paper):
S606(n) = S636(n-1)+ S636(n-2)+ S909(n-2)+ 3^(n-2)+ 2; n >= 2; S909(0)  = 0; S636(0)  = 0
Note: S636(n) and S909(n) are sequences A183116 and A183112 respectively.
Closed-Form Expression: Let
λ1 = [1+sqrt(26/27)]^(1/3) + [1-sqrt(26/27)]^(1/3)
λ2 = -0.5*λ1 + 0.5*i*{[sqrt(27)+sqrt(26)]^(1/3)- [sqrt(27)-sqrt(26)]^(1/3)}
λ3 = -0.5*λ1 - 0.5*i*{[sqrt(27)+sqrt(26)]^(1/3)- [sqrt(27)-sqrt(26)]^(1/3)}
AS = [(7/11)* λ2* λ3 - (10/11)*(λ2 + λ3) + (19/11)]/[(λ2 - λ1)*( λ3 - λ1)]
BS = [(7/11)* λ1* λ3 - (10/11)*(λ1 + λ3) + (19/11)]/[(λ1 - λ2)*( λ3 - λ2)]
CS = [(7/11)* λ1* λ2 - (10/11)*(λ1 + λ2) + (19/11)]/[(λ2 - λ3)*( λ1 - λ3)]
Then, for n > 0:
S606(n) = (10/33)*3^n + 0.5*AS*[(λ1 + 1)^2]*λ1^(n-1) + 0.5*BS*[(λ2 + 1)^2]*λ2^(n-1) + 0.5*CS*[(λ3 + 1)^2]*λ3^(n-1) - 2

cp's OEIS Frontend

A000244 Powers of 3: a(n) = 3^n.

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A003462 a(n) = (3^n - 1)/2.

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A183125 Magnetic Tower of Hanoi, total number of moves, generated by a certain algorithm, yielding a "forward moving" non-optimal solution of the [NEUTRAL ; NEUTRAL ; NEUTRAL] pre-colored puzzle.

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A122983 a(n) = (2 + (-1)^n + 3^n)/4.

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A104743 a(n) = 3^n + n.

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