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A007051
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(3^n + 1)/2
(Formerly M1458)
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119
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1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842, 29525, 88574, 265721, 797162, 2391485, 7174454, 21523361, 64570082, 193710245, 581130734, 1743392201, 5230176602, 15690529805, 47071589414, 141214768241, 423644304722, 1270932914165, 3812798742494, 11438396227481
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OFFSET
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0,2
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COMMENTS
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Number of ordered trees with n edges and height at most 4.
Number of palindromic structures using a maximum of three different symbols. - Marks R. Nester (nesterm(AT)dpi.qld.gov.au)
Number of compositions of all even natural numbers into n parts <=2 (0 is counted as a part), see example. [Adi Dani, May 14 2011]
Consider the mapping f(a/b) = (a + 2*b)/(2*a + b). Taking a = 1, b = 2 to start with, and carrying out this mapping repeatedly on each new (reduced) rational number gives the sequence 1/2, 4/5, 13/14, 40/41, ... converging to 1. The sequence contains the denominators = (3^n+1)/2. The same mapping for N i.e. f(a/b) = (a + N*b)/(a+b) gives fractions converging to N^(1/2). - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 22 2003
Second binomial transform of the expansion of cosh(x). - Paul Barry, Apr 05 2003
The sequence (1,1,2,5,..)=3^n/6+1/2+0^n/3 has binomial transform A007581. - Paul Barry, Jul 20 2003
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, Jun 10 2004
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
Sums of rows of the triangle in A119258. - Reinhard Zumkeller, May 11 2006
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
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, Jan 10 2008
a(n+1) gives the number of primitive periodic multiplex juggling sequences of length n with base state <2>. - Steve Butler, Jan 21 2008
a(n) is also the number of idempotent order-preserving and order-decreasing partial transformations (of an n-chain). [From Abdullahi Umar, Oct 02 2008]
Equals row sums of triangle A147292 [From Gary W. Adamson, Nov 05 2008]
Equals leftmost column of A071919^3 [From Gary W. Adamson, Apr 13 2009]
A010888(a(n))=5 for n>=2, that is, the digital root of the terms >=5 equals 5. [Parthasarathy Nambi, Jun 03 2009]
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=5, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=(-1)^n*charpoly(A,2). [From Milan Janjic, Jan 27 2010]
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=6, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=(-1)^(n-1)*charpoly(A,3). [From Milan Janjic, Feb 21 2010]
It appears that if s(n) is a rational sequence of the form s(1)=2, s(n)= (2*s(n-1)+1)/(s(n-1)+2),n>1 then s(n)=a(n)/(a(n-1)-1).
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REFERENCES
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J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 47.
S. Butler and R. Graham, Enumerating (multiplex) juggling sequences, arXiv:0801.2597
Adi Dani, Quasicompositions of natural numbers, Proceedings of III congress of mathematicians of Macedonia, Struga Macedonia 29 IX -2 X 2005 pages 225-238.
P. Duncan and Einar Steingrimsson, Pattern avoidance in ascent sequences, Arxiv preprint arXiv:1109.3641, 2011
Encyclopedia of Combinatorial Structures, Entry 454, divided by 2.
Hwang, F. K. and Mallows, C. L.; Enumerating nested and consecutive partitions. J. Combin. Theory Ser. A 70 (1995), no. 2, 323-333.
S. Kitaev, J. Remmel and M. Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, Arxiv preprint arXiv:1201.6243, 2012. - From N. J. A. Sloane, May 09 2012
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.
Kin Y. Li, Mathematical Excalibur, 4(1999) Number 4, p. 3, Problem 83
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.
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.
M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia.
L. Pudwell, Pattern avoidance in trees (slides from a talk, mentions many sequences), http://faculty.valpo.edu/lpudwell/slides/notredame.pdf, 2012. - From N. J. A. Sloane, Jan 03 2013
P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 60.
P. Ribenboim, The Little Book of Big Primes, Springer-Verlag, NY, 1991, p. 53.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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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 163
A. M. Hinz, S. Klavžar, U. Milutinović, C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 100. Book's website
Sergey Kitaev and Jeffrey Remmel, Simple marked mesh patterns, Arxiv preprint arXiv:1201.1323, 2012
Laradji, A. and Umar, A. Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra 278, (2004), 342-359.
Laradji, A. and Umar, A. Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq. 7 (2004), 04.3.8
Kin Y. Li, Problem 83
Nelma Moreira and Rogerio Reis, On the density of languages representing finite set partitions
Eric Weisstein's World of Mathematics, Mephisto Waltz Sequence
Index entries for sequences related to linear recurrences with constant coefficients, signature (4,-3).
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FORMULA
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a(n)=3*a(n-1)-1.
Binomial transform of Chebyshev coefficients A011782. - Paul Barry, Mar 16 2003
a(n)=4*a(n-1)-3*a(n-2), a(0)=1, a(1)=2. G.f.: (1-2*x)/((1-x)*(1-3*x)). - Paul Barry, Mar 16 2003
E.g.f.: exp(2*x)*cosh(x) - Paul Barry, Apr 05 2003
a(n)=sum{k=0..floor(n/2); C(n, 2*k)2^(n-2*k) } - Paul Barry, May 08 2003
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
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
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, Oct 15 2005
If p[i]=fibonacci(2i-3) 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 Janjic, May 08 2010]
INVERT transform of A001519: [1, 1, 2, 5, 13, 34,...]. - Gary W. Adamson, Jun 13 2011.
a(n) = M^n*[1,1,1,0,0,0,...], leftmost column term; where M = an infinite bidiagonal matrix with all 1's in the superdiagonal and (1,2,3,...) in the main diagonal and the rest zeros. - Gary W. Adamson, Jun 23, 2011
a(n) = M^n*{1,1,1,0,0,0,...], top entry term; where M is an infinite bidiagonal matrix with all 1's in the superdiagonal, (1,2,3,...) as the main diagonal, and the rest zeros. - Gary W. Adamson, Jun 24 2011
a(n) = A201730(n,0). - Philippe Deléham, Dec 05 2011
G.f.: G(0) where G(k)= 1 + 2*x/( 1-2*x - x*(1-2*x)/(x + (1-2*x)*2/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 10 2012
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EXAMPLE
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From Adi Dani, May 14 2011: (Start)
a(3)=14 because all compositions of even natural numbers into 3 parts <=2 are
for 0: (0,0,0)
for 2: (0,1,1), (1,0,1), (1,1,0), (0,0,2), (0,2,0), (2,0,0)
for 4: (0,2,2), (2,0.2), (2,2,0), (1,1,2), (1,2,1), (2,1,1)
for 6: (2,2,2).
(End)
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MAPLE
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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
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MATHEMATICA
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Table[(3^n + 1)/2, {n, 0, 50}] - Stefan Steinerberger, Apr 08 2006
a = 1; lst = {a}; Do[a = a*3 - 1; AppendTo[lst, a], {n, 0, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky_, Dec 25 2008 *)
CoefficientList[Series[(1 - 2 x)/((1 - x) (1 - 3 x)), {x, 0, 40}], x] (* From Harvey P. Dale, June 20 2011 *)
LinearRecurrence[{4, -3}, {2, 5}, {0, 28}] (* Arkadiusz Wesolowski, Oct 30 2012 *)
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PROG
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(PARI) a(n)=3^n>>1 \\ Charles R Greathouse IV, Jun 10 2011
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CROSSREFS
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Cf. A056449, A064881-A064886, A008277, A007581, A056272, A056273, A000392, A000079, A034472.
A147292 [From Gary W. Adamson, Nov 05 2008]
Cf. A003462 [From Vladimir Joseph Stephan Orlovsky, Dec 25 2008]
A071919 [From Gary W. Adamson, Apr 13 2009]
Sequence in context: A116849 A123183 * A124302 A088355 A113485 A054391
Adjacent sequences: A007048 A007049 A007050 * A007052 A007053 A007054
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KEYWORD
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easy,nonn,nice
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AUTHOR
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Colin Mallows, N. J. A. Sloane , Simon Plouffe, Robert G. Wilson v
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STATUS
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approved
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