%I M2847 #137 Mar 31 2024 17:25:25

%S 1,3,10,34,116,396,1352,4616,15760,53808,183712,627232,2141504,

%T 7311552,24963200,85229696,290992384,993510144,3392055808,11581202944,

%U 39540700160,135000394752,460920178688,1573679925248

%N Number of order-consecutive partitions of n.

%C After initial terms, first differs from A291292 at a(6) = 1352, A291292(8) = 1353.

%C 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".

%C It appears that this sequence is the BinomialMean transform of A001653 (see A075271). - _John W. Layman_, Oct 03 2002

%C 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

%C Equals the INVERT transform of (1, 2, 5, 13, 34, 89, ...). - _Gary W. Adamson_, May 01 2009

%C a(n) is the number of compositions of n when there are 3 types of ones. - _Milan Janjic_, Aug 13 2010

%C a(n)/a(n-1) tends to (4 + sqrt(8))/2 = 3.414213.... _Gary W. Adamson_, Jul 30 2013

%C a(n) is the first subdiagonal of array A228405. - _Richard R. Forberg_, Sep 02 2013

%C 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

%C From _Gus Wiseman_, Mar 05 2020: (Start)

%C 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:

%C (1) (1,1) (1,1,1)

%C (1,2) (1,1,2)

%C (2,1) (1,2,1)

%C (1,2,2)

%C (1,2,3)

%C (1,3,2)

%C (2,1,1)

%C (2,2,1)

%C (2,3,1)

%C (3,2,1)

%C Missing are: (2,1,2), (2,1,3), (3,1,2).

%C 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:

%C {{1}} {{1,2}} {{1,2,3}}

%C {{1},{2}} {{1},{2,3}}

%C {{2},{1}} {{1,2},{3}}

%C {{1,3},{2}}

%C {{2},{1,3}}

%C {{2,3},{1}}

%C {{3},{1,2}}

%C {{1},{2},{3}}

%C {{1},{3},{2}}

%C {{2},{1},{3}}

%C Cf. A000670, A056242, A332673, A332872. (End)

%C 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

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A007052/b007052.txt">Table of n, a(n) for n = 0..1000</a>

%H S. Barbero, U. Cerruti, and N. Murru, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barbero2/barbero7.html">A Generalization of the Binomial Interpolated Operator and its Action on Linear Recurrent Sequences </a>, J. Int. Seq. 13 (2010) # 10.9.7, proposition 16.

%H Jean-Luc Baril, José L. Ramírez, and Fabio A. Velandia, <a href="http://jl.baril.u-bourgogne.fr/hexbij.pdf">Bijections between Directed-Column Convex Polyominoes and Restricted Compositions</a>, September 29, 2023.

%H Tyler Clark and Tom Richmond, <a href="http://people.wku.edu/tom.richmond/Papers/CountConvexTopsFTOsets.pdf">The Number of Convex Topologies on a Finite Totally Ordered Set</a>, 2013, Involve, Vol. 8 (2015), No. 1, 25-32.

%H Pamela Fleischmann, Jonas Höfer, Annika Huch, and Dirk Nowotka, <a href="https://arxiv.org/abs/2306.14192">alpha-beta-Factorization and the Binary Case of Simon's Congruence</a>, arXiv:2306.14192 [math.CO], 2023.

%H Juan B. Gil and Jessica A. Tomasko, <a href="https://arxiv.org/abs/2108.06462">Fibonacci colored compositions and applications</a>, arXiv:2108.06462 [math.CO], 2021.

%H F. K. Hwang and C. L. Mallows, <a href="/A007051/a007051.pdf">Enumerating nested and consecutive partitions</a>, Preprint. (Annotated scanned copy)

%H F. K. Hwang and C. L. Mallows, <a href="http://dx.doi.org/10.1016/0097-3165(95)90097-7">Enumerating nested and consecutive partitions</a>, J. Combin. Theory Ser. A 70 (1995), no. 2, 323-333.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=164">Encyclopedia of Combinatorial Structures 164</a>

%H Mircea Merca, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL15/Merca1/merca6.html">A Note on Cosine Power Sums</a> J. Integer Sequences, Vol. 15 (2012), Article 12.5.3.

%H J.-C. Novelli and J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv preprint arXiv:1403.5962 [math.CO], 2014.

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H M. Z. Spivey and L. L. Steil, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.html"> The k-Binomial Transforms and the Hankel Transform</a>, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.

%H <a href="/index/Poi#poker">Index entries for sequences related to poker</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2).

%F a(n+1) = 4a(n) - 2a(n-1).

%F G.f.: (1-x)/(1-4x+2x^2).

%F Binomial transform of Pell numbers 1, 2, 5, 12, ... (A000129).

%F a(n) = A006012(n+1)/2 = A056236(n+1)/4. - _Michael Somos_, Mar 06 2003

%F 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

%F 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

%F a(n) = ( (2-sqrt(2))^(n+1) + (2+sqrt(2))^(n+1) )/4. - _Herbert Kociemba_, Jun 12 2004

%F 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

%F 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

%F E.g.f. : exp(2x)(cosh(sqrt(2x)+sinh(sqrt(2)x)/sqrt(2). - _Paul Barry_, Nov 20 2003

%F a(n) = A007068(2*n), n>0. - _R. J. Mathar_, Aug 17 2009

%F 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

%F 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

%F 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

%F a(n) = 3*a(n-1) + a(n-2) + a(n-3) + a(n-4) + ... + a(0). - _Gary W. Adamson_, Aug 12 2013

%F a(n) = a(-2-n) * 2^(n+1) for all n in Z. - _Michael Somos_, Jan 25 2017

%e G.f. = 1 + 3*x + 10*x^2 + 34*x^3 + 116*x^4 + 396*x^5 + 1352*x^6 + 4616*x^7 + ...

%t 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 *)

%t a[ n_] := ((2 + Sqrt[2])^(n + 1) + (2 - Sqrt[2])^(n + 1)) / 4 // Simplify; (* _Michael Somos_, Jan 25 2017 *)

%t LinearRecurrence[{4, -2}, {1, 3}, 24] (* _Jean-François Alcover_, Jan 07 2019 *)

%t unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];

%t allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];

%t Table[Length[Select[Union@@Permutations/@allnorm[n],unimodQ]],{n,6}] (* _Gus Wiseman_, Mar 06 2020 *)

%o (PARI) {a(n) = real((2 + quadgen(8))^(n+1)) / 2}; /* _Michael Somos_, Mar 06 2003 */

%o (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

%Y Cf. A006012, A003480, A056236.

%Y First differences of A007070.

%Y Cf. A000129, A000670, A001523, A001653, A007068, A035344, A060223, A075271, A227038, A291292, A328509, A332577, A332743, A332873.

%K nonn,easy

%O 0,2

%A Colin Mallows, _N. J. A. Sloane_, _Simon Plouffe_