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COMMENTS
| 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 (kociemba(AT)t-online.de), Jun 10 2004
a(1) = 1, a(n+1) is the least number so 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 (amarnath_murthy(AT)yahoo.com), Nov 28 2004
a(n) = sum of (n+1)-th row in Triangle A105728. - Reinhard Zumkeller, Apr 18 2005
With p(n) = the number of integer partitions of n, p(i) = the number of parts of the i-th partition of n, d(i) = the number of different parts of the i-th partition of n, m(i,j) = multiplicity of the j-th part of the i-th partition of n, sum_{i=1}^{p(n)} = sum over i and prod_{j=1}^{d(i)} = product over j one has: a(n)=sum_{i=1}^{p(n)} (p(i)!/(prod_{j=1}^{d(i)} m(i,j)!))*2^(p(i)-1) - Thomas Wieder (wieder.thomas(AT)t-online.de), 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 (rlahaye(AT)new.rr.com), Dec 22 2006
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 R. Janjic (agnus(AT)blic.net), May 24 2007
1/1 + 1/3 + 1/9 + ... = 3/2 [From Gary W. Adamson, Aug 29 2008]
Equals row sums of triangle A125076 [From Gary W. Adamson, Dec 18 2008]
Equals row sums of triangle A153279 [From Gary W. Adamson, Dec 23 2008]
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. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Feb 28 2009]
a(n) = A064614(A000079(n)) and A064614(m)<a(n) for m < A000079(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 08 2010]
Contribution from Gary W. Adamson, May 17 2010: (Start)
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) (End)
a(n) is the number of generalized compositions of n when there are 3*2^i different types of i, (i=1,2,...). [From Milan R. Janjic (agnus(AT)blic.net), 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,...). [From Milan R. Janjic (agnus(AT)blic.net), 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. [From Juri-Stepan Gerasimov, Mar 17 2011]
Sum of coefficients of the expansion of (1+x+x^2)^n. [From 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). [From 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
Also, first and least element of the matrix [1,sqrt(2); sqrt(2),2]^(n+1). - M. F. Hasler, Nov 25 2011.
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FORMULA
| a(n) = 3^n.
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 (benoit7848c(AT)orange.fr), Nov 01 2002
a(n) = Sum_{k=0..n} 2^k*binomial(n, k).
a(n) = A090888(n, 2). - Ross La Haye (rlahaye(AT)new.rr.com), Sep 21 2004
a(n) = 2^(2n) - A005061(n). - Ross La Haye (rlahaye(AT)new.rr.com), Sep 10 2005
a(n) = A112626(n, 0). - Ross La Haye (rlahaye(AT)new.rr.com), Jan 11 2006
Hankel transform of A007854 = [1, 3, 12, 51, 222, 978, 4338, ...] . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 26 2006
Binomial transform of the powers of two: (1, 2, 4, 8,...). - Gary W. Adamson, Sep 20 2007
a(n) = 2*StirlingS2(n+1,3) + StirlingS2(n+2,2) = 2*(StirlingS2(n+1,3) + StirlingS2(n+1,2)) + 1. - Ross La Haye (rlahaye(AT)new.rr.com), 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 (rlahaye(AT)new.rr.com), Jun 09 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. [From Milan R. Janjic (agnus(AT)blic.net), May 08 2010]
G.f. A(x)=M(x)/(1-M(x))^2, M(x) - o.g.f for Motzkin numbers (A0001006) [From Kruchinin Vladimir, 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]
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