Search: id:A000244 Results 1-1 of 1 results found. %I A000244 M2807 N1129 %S A000244 1,3,9,27,81,243,729,2187,6561,19683,59049,177147,531441,1594323,4782969, %T A000244 14348907,43046721,129140163,387420489,1162261467,3486784401,10460353203, %U A000244 31381059609,94143178827,282429536481,847288609443,2541865828329,7625597484987 %N A000244 Powers of 3. %C A000244 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 A000244 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 %C A000244 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 %C A000244 a(n) = sum of (n+1)-th row in Triangle A105728. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 18 2005 %C A000244 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 %C A000244 a(n) = A112626(n, 0). - Ross La Haye (rlahaye(AT)new.rr.com), Jan 11 2006 %C A000244 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 %C A000244 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 %C A000244 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 %C A000244 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 %C A000244 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) %C A000244 1/1 + 1/3 + 1/9 + ... = 3/2 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 29 2008] %C A000244 Equals row sums of triangle A125076 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 18 2008] %C A000244 Equals row sums of triangle A153279 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 23 2008] %C A000244 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] %D A000244 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] %D A000244 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). %D A000244 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %H A000244 T. D. Noe, Table of n, a(n) for n = 0..200 %H A000244 P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5. %H A000244 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 7 %H A000244 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 268 %H A000244 Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets %H A000244 Tanya Khovanova, Recursive Sequences %H A000244 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. %H A000244 S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992. %H A000244 Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1. %H A000244 Eric Weisstein's World of Mathematics, Hanoi Graph %H A000244 Eric Weisstein's World of Mathematics, Sierpinski Graph %H A000244 Index entries for "core" sequences %H A000244 Index entries for sequences related to linear recurrences with constant coefficients %H A000244 Index entries for related partition-counting sequences %F A000244 a(n) = 3^n; a(n) = 3*a(n-1). %F A000244 G.f.: 1/(1-3x), e.g.f.: exp(3x) %F A000244 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 %F A000244 3^n = Sum_{k=0..n} 2^k*binomial(n, k). %F A000244 a(n) = A090888(n, 2). - Ross La Haye (rlahaye(AT)new.rr.com), Sep 21 2004 %F A000244 a(n) = 2^(2n) - A005061(n). - Ross La Haye (rlahaye(AT)new.rr.com), Sep 10 2005 %F A000244 Hankel transform of A007854 = [1, 3, 12, 51, 222, 978, 4338, ...] . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 26 2006 %F A000244 Binomial transform of the powers of two: (1, 2, 4, 8,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 20 2007 %F A000244 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 %F A000244 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 %p A000244 A000244 := n->3^n; [ seq(3^n,n=0..50) ]; %p A000244 A000244:=-1/(-1+3*z); [S. Plouffe in his 1992 dissertation.] %p A000244 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 %p A000244 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] %p A000244 with(finance):seq(futurevalue(3,2,n), n=-1..26);# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 25 2009] %p A000244 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] %t A000244 Table[3^n, {n, 0, 30}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 01 2006 %t A000244 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] %o A000244 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 %o A000244 (PARI) A000244(n) = 3^n [From Michael Porter (michael_b_porter(AT)yahoo.com), Nov 03 2009] %Y A000244 a(n) = A092477(n, 2) for n>0. %Y A000244 Cf. A100772. %Y A000244 A125076 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 18 2008] %Y A000244 A153279 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 23 2008] %Y A000244 a(n) = A159991(n)/A009964(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 02 2009] %Y A000244 Sequence in context: A133494 A140429 A141413 this_sequence A050733 A079846 A067500 %Y A000244 Adjacent sequences: A000241 A000242 A000243 this_sequence A000245 A000246 A000247 %K A000244 nice,nonn,easy,core,new %O A000244 0,2 %A A000244 N. J. A. Sloane (njas(AT)research.att.com). Search completed in 0.004 seconds