%I A007051 M1458
%S A007051 1,2,5,14,41,122,365,1094,3281,9842,29525,88574,265721,797162,
%T A007051 2391485,7174454,21523361,64570082,193710245,581130734,1743392201,
%U A007051 5230176602,15690529805,47071589414,141214768241,423644304722
%N A007051 (3^n + 1)/2.
%C A007051 Number of ordered trees with n edges and height at most 4.
%C A007051 Number of palindromic structures using a maximum of three different symbols. 
               - Marks R. Nester (nesterm(AT)dpi.qld.gov.au)
%C A007051 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 following sequence 1/2,4/5,13/14,40/41,
               ... converging to 1. Sequence contains the denominators = (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 (amarnath_murthy(AT)yahoo.com), 
               Mar 22 2003
%C A007051 Second binomial transform of expansion of cosh(x). - Paul Barry (pbarry(AT)wit.ie), 
               Apr 05 2003
%C A007051 The sequence (1,1,2,5,..)=3^n/6+1/2+0^n/3 has binomial transform A007581. 
               - Paul Barry (pbarry(AT)wit.ie), Jul 20 2003
%C A007051 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) = 1. - Herbert 
               Kociemba (kociemba(AT)t-online.de), Jun 10 2004
%C A007051 Density of regular language L over {1,2,3}^* (i.e. number of strings 
               of length n in L) described by regular expression 11*+11*2(1+2)*+11*2(1+2)*3(1+2+3)*. 
               - Nelma Moreira (nam(AT)ncc.up.pt), Oct 10 2004
%C A007051 Sums of rows of the triangle in A119258. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               May 11 2006
%C A007051 Number of n-words from the alphabet A={a,b,c} which contain an even number 
               of a's. - Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Aug 
               30 2006
%C A007051 Let P(A) be the power set of an n-element set A. Then a(n) = the number 
               of pairs of elements {x,y} of P(A) for which either 0) x and y are 
               disjoint and for which x is not a subset of y and y is not a subset 
               of x, or 1) x = y. - Ross La Haye (rlahaye(AT)new.rr.com), Jan 10 
               2008
%C A007051 a(n+1) gives the number of primitive periodic multiplex juggling sequences 
               of length n with base state <2>. - Steve Butler (sbutler(AT)math.ucsd.edu), 
               Jan 21 2008
%C A007051 a(n) is also the number of idempotent order-preserving and order-decreasing 
               partial transformations (of an n-chain). [From A. Umar (aumarh(AT)squ.edu.om), 
               Oct 02 2008]
%C A007051 Equals row sums of triangle A147292 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Nov 05 2008]
%C A007051 Equals leftmost column of A071919^3 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Apr 13 2009]
%C A007051 Contribution from Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Jun 
               03 2009: (Start)
%C A007051 Repetitive digit sum of terms in the sequence gives 5. Here are few examples:
%C A007051 47071589414 is a term in this sequence, whose repetitive digit sum is: 
               47071589414 --> 50 --> 5
%C A007051 141214768241 is a term in this sequence, whose repetitive digit sum is: 
               141214768241 --> 41 --> 5
%C A007051 423644304722 is a term in this sequence, whose repetitive digit sum is: 
               423644304722 --> 41 --> 5
%C A007051 (End)
%D A007051 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A007051 Hwang, F. K. and Mallows, C. L.; Enumerating nested and consecutive partitions. 
               J. Combin. Theory Ser. A 70 (1995), no. 2, 323-333.
%D A007051 M. R. Nester (1999). Mathematical investigations of some plant interaction 
               designs. PhD Thesis. University of Queensland, Brisbane, Australia.
%D A007051 P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 
               2nd ed., 1989, p. 60.
%D A007051 P. Ribenboim, The Little Book of Big Primes, Springer-Verlag, NY, 1991, 
               p. 53.
%D A007051 Nelma Moreira and Rogerio Reis, On the density of languages representing 
               finite set partitions, Technical Report DCC-2004-07, August 2004, 
               DCC-FC& LIACC, Universidade do Porto.
%D A007051 Kin Y. Li, Mathematical Excalibur, 4(1999) Number 4, p. 3, Problem 83
%D A007051 N. Moreira and R. Reis, On the Density of Languages Representing Finite 
               Set Partitions, Journal of Integer Sequences, Vol. 8 (2005), Article 
               05.2.8.
%D A007051 S. Butler and R. Graham, Enumerating (multiplex) juggling sequences, 
               arXiv:0801.2597
%D A007051 Encyclopedia of Combinatorial Structures, Entry 454, divided by 2.
%D A007051 Laradji, A. and Umar, A. Combinatorial results for semigroups of order-decreasing 
               partial transformations J. Integer Seq. 7 (2004), 04.3.8, 14 pp. 
               [From A. Umar (aumarh(AT)squ.edu.om), Oct 02 2008]
%D A007051 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]
%H A007051 T. D. Noe, <a href="b007051.txt">Table of n, a(n) for n=0..200</a>
%H A007051 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A007051 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Sequences realized by oligomorphic permutation groups</a>, J. Integ. 
               Seqs. Vol. 3 (2000), #00.1.5.
%H A007051 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=163">
               Encyclopedia of Combinatorial Structures 163</a>
%H A007051 Nelma Moreira and Rogerio Reis, <a href="http://www.dcc.fc.up.pt/Pubs/
               TR04/dcc-2004-07.ps.gz">dcc-2004-07.ps</a>
%H A007051 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               MephistoWaltzSequence.html">Mephisto Waltz Sequence</a>
%H A007051 Kin Y. Li, <a href="http://www.math.ust.hk/excalibur/v4_n4.pdf">Problem 
               83</a>
%H A007051 Laradji, A. and Umar, A., <a href="http://www.cs.uwaterloo.ca/journals/
               JIS/">Combinatorial Results for Semigroups of Order-Decreasing Partial 
               Transformations </a>, Journal of Integer Sequences, Vol. 7 (2004), 
               Article 04.3.8. [From A. Umar (aumarh(AT)squ.edu.om), Oct 02 2008]
%F A007051 a(n)=3*a(n-1)-1.
%F A007051 Binomial transform of Chebyshev coefficients A011782. - Paul Barry (pbarry(AT)wit.ie), 
               Mar 16 2003
%F A007051 a(n)=4a(n-1)-3a(n-2), a(0)=1, a(1)=2. G.f.: (1-2x)/((1-x)(1-3x)). - Paul 
               Barry (pbarry(AT)wit.ie), Mar 16 2003
%F A007051 E.g.f. exp(2x)cosh(x) - Paul Barry (pbarry(AT)wit.ie), Apr 05 2003
%F A007051 a(n)=sum{k=0..floor(n/2); C(n, 2k)2^(n-2k) } - Paul Barry (pbarry(AT)wit.ie), 
               May 08 2003
%F A007051 This sequence is also the partial sums of the first 3 Stirling numbers 
               of second kind: a(n) = S(n+1, 1)+S(n+1, 2)+S(n+1, 3) for n>=0; alternatively 
               it is the number of partitions of [n+1] into 3 or fewer parts. - 
               Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Jun 21 2004
%F A007051 For c=3, a(n)= (c^n)/c!+sum_{k=1..c-2}((k^n)/k!*(sum_{j=2..c-k}(((-1)^j)/
               j!))) or = sum_{k=1..c}(g(k, c)*k^n) where g(1, 1)=1 g(1, c)=g(1, 
               c-1)+((-1)^(c-1))/(c-1)!, c>1 g(k, c)=g(k-1, c-1)/k, for c>1 and 
               2<= k<= c - Nelma Moreira (nam(AT)ncc.up.pt), Oct 10 2004
%F A007051 The i-th term of the sequence is the entry (1, 1) in the i-th power of 
               the 2 by 2 matrix M=((2, 1), (1, 2)). - Simone Severini (ss54(AT)york.ac.uk), 
               Oct 15 2005
%p A007051 with (combinat):seq(sum(stirling2(n, j),j=1..3), n=1..26); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Dec 04 2007
%p A007051 ZL := [S, {S=Union(Sequence(Z), Sequence(Union(Z, Z, Z)))}, unlabeled]: 
               seq(combstruct[count](ZL, size=n)/2, n=0..25); - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Jun 19 2008
%t A007051 Table[(3^n + 1)/2, {n, 0, 50}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), 
               Apr 08 2006
%t A007051 a=1;lst={a};Do[a=a*3-1;AppendTo[lst,a],{n,0,5!}];lst [From Vladimir Orlovsky 
               (4vladimir(AT)gmail.com), Dec 25 2008]
%o A007051 sage: [lucas_number2(n,4,3)/2 for n in xrange(0,33)] - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Jul 08 2008
%o A007051 (Other) sage: [(sigma(3,n-1))/2 for n in xrange(1,27)] # [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Jun 05 2009]
%Y A007051 Cf. A056449, A064881-A064886.
%Y A007051 Cf. A008277.
%Y A007051 Cf. A007581, A056272, A056273.
%Y A007051 Cf. A000392, A000079.
%Y A007051 Cf. A034472.
%Y A007051 A147292 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 05 2008]
%Y A007051 Cf. A003462 [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 25 
               2008]
%Y A007051 A071919 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 13 2009]
%Y A007051 Sequence in context: A122055 A116845 A116849 this_sequence A124302 A123183 
               A088355
%Y A007051 Adjacent sequences: A007048 A007049 A007050 this_sequence A007052 A007053 
               A007054
%K A007051 easy,nonn,nice
%O A007051 0,2
%A A007051 Colin Mallows, N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe, 
               Robert G. Wilson v (rgwv(AT)rgwv.com)