OFFSET
0,2
COMMENTS
Joe Keane (jgk(AT)jgk.org) observes that this sequence (beginning at 3) is "size of raises in pot-limit poker, one blind, maximum raising".
It appears that this sequence is the BinomialMean transform of A001653 (see A075271). - John W. Layman, Oct 03 2002
Number of (s(0), s(1), ..., s(2n+1)) such that 0 < s(i) < 8 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+1, s(0) = 3, s(2n+1) = 4. - Herbert Kociemba, Jun 12 2004
Equals the INVERT transform of (1, 2, 5, 13, 34, 89, ...). - Gary W. Adamson, May 01 2009
a(n) is the number of compositions of n when there are 3 types of ones. - Milan Janjic, Aug 13 2010
a(n)/a(n-1) tends to (4 + sqrt(8))/2 = 3.414213.... Gary W. Adamson, Jul 30 2013
a(n) is the first subdiagonal of array A228405. - Richard R. Forberg, Sep 02 2013
Number of words of length n over {0,1,2,3,4} in which binary subwords appear in the form 10...0. - Milan Janjic, Jan 25 2017
From Gus Wiseman, Mar 05 2020: (Start)
Also the number of unimodal sequences of length n + 1 covering an initial interval of positive integers, where a sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence. For example, the a(0) = 1 through a(2) = 10 sequences are:
(1) (1,1) (1,1,1)
(1,2) (1,1,2)
(2,1) (1,2,1)
(1,2,2)
(1,2,3)
(1,3,2)
(2,1,1)
(2,2,1)
(2,3,1)
(3,2,1)
Missing are: (2,1,2), (2,1,3), (3,1,2).
Conjecture: Also the number of ordered set partitions of {1..n + 1} where no element of any block is greater than any element of a non-adjacent consecutive block. For example, the a(0) = 1 through a(2) = 10 ordered set partitions are:
{{1}} {{1,2}} {{1,2,3}}
{{1},{2}} {{1},{2,3}}
{{2},{1}} {{1,2},{3}}
{{1,3},{2}}
{{2},{1,3}}
{{2,3},{1}}
{{3},{1,2}}
{{1},{2},{3}}
{{1},{3},{2}}
{{2},{1},{3}}
a(n-1) is the number of hexagonal directed-column convex polyominoes having area n (see Baril et al. at page 4). - Stefano Spezia, Oct 14 2023
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
S. Barbero, U. Cerruti, and N. Murru, A Generalization of the Binomial Interpolated Operator and its Action on Linear Recurrent Sequences , J. Int. Seq. 13 (2010) # 10.9.7, proposition 16.
Jean-Luc Baril, José L. Ramírez, and Fabio A. Velandia, Bijections between Directed-Column Convex Polyominoes and Restricted Compositions, September 29, 2023.
Tyler Clark and Tom Richmond, The Number of Convex Topologies on a Finite Totally Ordered Set, 2013, Involve, Vol. 8 (2015), No. 1, 25-32.
Pamela Fleischmann, Jonas Höfer, Annika Huch, and Dirk Nowotka, alpha-beta-Factorization and the Binary Case of Simon's Congruence, arXiv:2306.14192 [math.CO], 2023.
Juan B. Gil and Jessica A. Tomasko, Fibonacci colored compositions and applications, arXiv:2108.06462 [math.CO], 2021.
F. K. Hwang and C. L. Mallows, Enumerating nested and consecutive partitions, Preprint. (Annotated scanned copy)
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 164
Mircea Merca, A Note on Cosine Power Sums J. Integer Sequences, Vol. 15 (2012), Article 12.5.3.
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
N. J. A. Sloane, Transforms
M. Z. Spivey and L. L. Steil, The k-Binomial Transforms and the Hankel Transform, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.
Index entries for linear recurrences with constant coefficients, signature (4,-2).
FORMULA
a(n+1) = 4*a(n) - 2*a(n-1).
G.f.: (1-x)/(1-4*x+2*x^2).
Binomial transform of Pell numbers 1, 2, 5, 12, ... (A000129).
a(n) = (A035344(n)+1)/2; a(n) = (2+sqrt(2))^n(1/2+sqrt(2)/4)+(2-sqrt(2))^n(1/2-sqrt(2)/4). - Paul Barry, Jul 16 2003
Second binomial transform of (1, 1, 2, 2, 4, 4, ...). a(n) = Sum_{k=1..floor(n/2)}, C(n, 2k)*2^(n-k-1). - Paul Barry, Nov 22 2003
a(n) = ( (2-sqrt(2))^(n+1) + (2+sqrt(2))^(n+1) )/4. - Herbert Kociemba, Jun 12 2004
a(n) = both left and right terms in M^n * [1 1 1], where M = the 3 X 3 matrix [1 1 1 / 1 2 1 / 1 1 1]. M^n * [1 1 1] = [a(n) A007070(n) a(n)]. E.g., a(3) = 34. M^3 * [1 1 1] = [34 48 34] (center term is A007070(3)). - Gary W. Adamson, Dec 18 2004
The i-th term of the sequence is the entry (2, 2) in the i-th power of the 2 X 2 matrix M = ((1, 1), (1, 3)). - Simone Severini, Oct 15 2005
E.g.f.: exp(2*x)*(cosh(sqrt(2)*x)+sinh(sqrt(2)*x)/sqrt(2)). - Paul Barry, Nov 20 2003
a(n) = A007068(2*n), n>0. - R. J. Mathar, Aug 17 2009
If p[i]=Fibonacci(2i-1) 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
a(n-1) = Sum_{k=-floor(n/4)..floor(n/4)} (-1)^k*binomial(2*n,n+4*k)/2. - Mircea Merca, Jan 28 2012
G.f.: G(0)*(1-x)/(2*x) + 1 - 1/x, where G(k) = 1 + 1/(1 - x*(2*k-1)/(x*(2*k+1) - (1-x)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013
a(n) = 3*a(n-1) + a(n-2) + a(n-3) + a(n-4) + ... + a(0). - Gary W. Adamson, Aug 12 2013
a(n) = a(-2-n) * 2^(n+1) for all n in Z. - Michael Somos, Jan 25 2017
EXAMPLE
G.f. = 1 + 3*x + 10*x^2 + 34*x^3 + 116*x^4 + 396*x^5 + 1352*x^6 + 4616*x^7 + ...
MATHEMATICA
a[n_]:=(MatrixPower[{{3, 1}, {1, 1}}, n].{{2}, {1}})[[2, 1]]; Table[a[n], {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
a[ n_] := ((2 + Sqrt[2])^(n + 1) + (2 - Sqrt[2])^(n + 1)) / 4 // Simplify; (* Michael Somos, Jan 25 2017 *)
LinearRecurrence[{4, -2}, {1, 3}, 24] (* Jean-François Alcover, Jan 07 2019 *)
unimodQ[q_]:=Or[Length[q]<=1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]];
allnorm[n_]:=If[n<=0, {{}}, Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Union@@Permutations/@allnorm[n], unimodQ]], {n, 6}] (* Gus Wiseman, Mar 06 2020 *)
PROG
(PARI) {a(n) = real((2 + quadgen(8))^(n+1)) / 2}; /* Michael Somos, Mar 06 2003 */
(Magma) [Floor((2+Sqrt(2))^n*(1/2+Sqrt(2)/4)+(2-Sqrt(2))^n*(1/2-Sqrt(2)/4)): n in [0..30] ] ; // Vincenzo Librandi, Aug 20 2011
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
Colin Mallows, N. J. A. Sloane, Simon Plouffe
STATUS
approved