0,2

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 (reinhard.zumkeller(AT)gmail.com), 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

a(n) = A112626(n, 0). - Ross La Haye (rlahaye(AT)new.rr.com), Jan 11 2006

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 (blekraj(AT)yahoo.com), 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 (FrankTAW(AT)Netscape.net), 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

Number of n-permutations of 4 objects u, v, z, x with repetition allowed and containing no u's. Permutations with repetitions. If n=1, then 3^1=3, >> v, z, x. (no u's) If n=2, then 3^2=9, >> vv, vz, vx, zz, zv, zx, xx, xv, xz,(no u's) If n=3, then 3^3=27, >> vvv, zzz, xxx, vvz, vzv, zvv, vzz, zvz, zzv, vvx, vxv, xvv, vxx, xvx, xxv, zzx, zxz, xzz, zxx, xxz, xxz, vxz, xzv, vzx, xvz, zvx, zxv. (no u's)

1/1 + 1/3 + 1/9 + ... = 3/2 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 29 2008]

Equals row sums of triangle A125076 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 18 2008]

Equals row sums of triangle A153279 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 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 funtions 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]

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. [From Ross La Haye (rlahaye(AT)new.rr.com), Feb 22 2009]

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

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 7

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 268

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Tanya Khovanova, Recursive Sequences

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.

Eric Weisstein's World of Mathematics, Hanoi Graph

Eric Weisstein's World of Mathematics, Sierpinski Graph

Index entries for "core" sequences

Index entries for sequences related to linear recurrences with constant coefficients

Index entries for related partition-counting sequences

a(n) = 3^n; a(n) = 3*a(n-1).

G.f.: 1/(1-3x), e.g.f.: exp(3x)

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

3^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

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 (qntmpkt(AT)yahoo.com), 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

A000244 := n->3^n; [ seq(3^n, n=0..50) ];

A000244:=-1/(-1+3*z); [S. Plouffe in his 1992 dissertation.]

a[0]:=1:a[1]:=3:for n from 2 to 50 do a[n]:=2*a[n-1]+3*a[n-2] od: seq(a[n], n=0..27); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 28 2008

a:=n->mul(2+mul(1, j=1..n), j=1..n):seq(a(n), n=0..32); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 06 2008]

with(finance):seq(futurevalue(3, 2, n), n=-1..26); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 25 2009]

with(combstruct): SeqSeqSeqL := [T, {T=Sequence(S, card > 0), S=Sequence(U, card > 0), U=Sequence(Z, card >0)}, unlabeled]: seq(count(SeqSeqSeqL, size=j), j=1..28); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 04 2009]

Table[3^n, {n, 0, 30}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 01 2006

aa = {}; Do[If[2 n - EulerPhi[6 n] == 0, AppendTo[aa, n]], {n, 1, 2187}]; bb={}; aa = {}; Do[If[2 n - EulerPhi[6 n] == 0, AppendTo[bb, EulerPhi[n]]], {n, 1, 100000}]; Union[bb]; Complement[aa, bb] [From Artur Jasinski (grafix(AT)csl.pl), Nov 06 2008]

sage: from sage.combinat.sloane_functions import recur_gen2b sage: it =recur_gen2b(1, n/18, n/18, 0, lambda n: 0) sage: [it.next() for i in range(29)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 16 2008

(PARI) A000244(n) = 3^n [From Michael Porter (michael_b_porter(AT)yahoo.com), Nov 03 2009]

a(n) = A092477(n, 2) for n>0.

Cf. A100772.

A125076 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 18 2008]

A153279 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 23 2008]

a(n) = A159991(n)/A009964(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 02 2009]

Sequence in context: A133494 A140429 A141413 this_sequence A050733 A079846 A067500

Adjacent sequences: A000241 A000242 A000243 this_sequence A000245 A000246 A000247

nice,nonn,easy,core,new

N. J. A. Sloane (njas(AT)research.att.com).