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A007051 (3^n + 1)/2.
(Formerly M1458)
139
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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, 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). - Abdullahi Umar, Oct 02 2008

Equals row sums of triangle A147292. - Gary W. Adamson, Nov 05 2008

Equals leftmost column of A071919^3. - 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). - 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). - 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).

Form an array with m(1,n)=1 and m(i,j)=sum[m(k,j){k=1..i-1}] + sum[m(i,k) {k=1..j-1}], which is the sum of the terms to the left of m(i,j) plus the sum above m(i,j).  The sum of the terms in antidiagonal(n-1)=a(n). - J. M. Bergot, Jul 16 2013

From Peter Bala, Oct 29 2013: (Start)

An Engel expansion of 3 to the base b := 3/2 as defined in A181565, with the associated series expansion 3 = b + b^2/2 + b^3/(2*5) + b^4/(2*5*14) + .... Cf. A034472.

More generally, for a positive integer n >= 3, the sequence [1, n - 1, n^2 - n - 1, ..., ( (n - 2)*n^k + 1 )/(n - 1), ...] is an Engel expansion of n/(n - 2) to the base n/(n - 1). Cases include A007583 (n = 4), A083065 (n = 5) and A083066 (n = 6).  (End)

Diagonal elements (and one more than anti-diagonal elements) of the matrix A^n where A=(2,1;1,2). - David Neil McGrath, Aug 17 2014

REFERENCES

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.

Adi Dani, Quasicompositions of natural numbers, Proceedings of III congress of mathematicians of Macedonia, Struga Macedonia 29 IX -2 X 2005 pages 225-238.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia.

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).

LINKS

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

A. M. Baxter, L. K. Pudwell, Ascent sequences avoiding pairs of patterns, 2014.

S. Butler and R. Graham, Enumerating (multiplex) juggling sequences, arXiv:0801.2597

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

F. Castro-Velez, A. Diaz-Lopez, R. Orellana, J. Pastrana and R. Zevallos, Number of permutations with same peak set for signed permutations, arXiv preprint arXiv: 1308.6621, 2013

P. Duncan and Einar Steingrimsson, Pattern avoidance in ascent sequences, Arxiv preprint arXiv:1109.3641, 2011

F.K Hwang and C.L Mallows, Enumerating nested and consecutive partitions, J. Combin. Theory Ser. A 70 (1995), no. 2, 323-333.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 163

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 454, divided by 2.

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

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.

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, Mathematical Excalibur, 4(1999) Number 4, p. 3.

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.

D. Necas, I. Ohlidal, Consolidated series for efficient calculation of the reflection and transmission in rough multilayers, Optics Express, Vol. 22, 2014, No. 4; DOI:10.1364/OE.22.004499. See Table 1.

L. Pudwell, Pattern avoidance in trees, (slides from a talk, mentions many sequences), 2012. - From N. J. A. Sloane, Jan 03 2013

Eric Weisstein's World of Mathematics, Mephisto Waltz Sequence

Index entries for sequences related to linear recurrences with constant coefficients, signature (4,-3).

FORMULA

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, 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. - 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

EXAMPLE

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)

MAPLE

ZL := [S, {S=Union(Sequence(Z), Sequence(Union(Z, Z, Z)))}, unlabeled]: seq(combstruct[count](ZL, size=n)/2, n=0..25); # Zerinvary Lajos, Jun 19 2008

MATHEMATICA

Table[(3^n + 1)/2, {n, 0, 50}] (* Stefan Steinerberger, Apr 08 2006 *)

CoefficientList[Series[(1 - 2 x)/((1 - x) (1 - 3 x)), {x, 0, 40}], x] (* Harvey P. Dale, Jun 20 2011 *)

LinearRecurrence[{4, -3}, {2, 5}, {0, 28}] (* Arkadiusz Wesolowski, Oct 30 2012 *)

PROG

(PARI) a(n)=3^n>>1 \\ Charles R Greathouse IV, Jun 10 2011

CROSSREFS

Cf. A056449, A064881-A064886, A008277, A007581, A056272, A056273, A000392, A000079, A034472.

Cf. A147292. - Gary W. Adamson, Nov 05 2008

Cf. A003462. - Vladimir Joseph Stephan Orlovsky, Dec 25 2008

Cf. A071919. - Gary W. Adamson, Apr 13 2009

Cf. A007583, A034472, A083065, A083066.

Sequence in context: A116849 A123183 * A124302 A088355 A113485 A054391

Adjacent sequences:  A007048 A007049 A007050 * A007052 A007053 A007054

KEYWORD

easy,nonn,nice

AUTHOR

Colin Mallows, N. J. A. Sloane, Simon Plouffe, Robert G. Wilson v

STATUS

approved

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Last modified September 22 04:02 EDT 2014. Contains 247035 sequences.