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)lhsystems.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

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.

T. D. Noe, Table of n, a(n) for n = 0..200

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

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.

Tanya Khovanova, Recursive Sequences

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

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

Index entries for "core" sequences

Index entries for related partition-counting sequences

Eric Weisstein's World of Mathematics, Hanoi Graph

Eric Weisstein's World of Mathematics, Sierpinski Graph

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

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

A000244:=-1/(-1+3*z); [Conjectured by 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

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

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

Cf. A100772.

Adjacent sequences: A000241 A000242 A000243 this_sequence A000245 A000246 A000247

Sequence in context: A022014 A038002 A133494 this_sequence A050733 A079846 A067500

nice,nonn,easy,core

njas