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A007052
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Number of order-consecutive partitions of n.
(Formerly M2847)
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87
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1, 3, 10, 34, 116, 396, 1352, 4616, 15760, 53808, 183712, 627232, 2141504, 7311552, 24963200, 85229696, 290992384, 993510144, 3392055808, 11581202944, 39540700160, 135000394752, 460920178688, 1573679925248
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OFFSET
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0,2
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COMMENTS
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After initial terms, first differs from A291292 at a(6) = 1352, A291292(8) = 1353.
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 sub-diagonal 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}}
Cf. A000670, A056242, A332673, A332872.
(End)
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REFERENCES
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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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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
S. Barbero, U. Cerruti, 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.
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.
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, 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 sequences related to poker
Index entries for linear recurrences with constant coefficients, signature (4,-2).
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FORMULA
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a(n+1) = 4a(n)-2a(n-1). G.f.: (1-x)/(1-4x+2x^2). Binomial transform of Pell numbers 1, 2, 5, 12, ... (A000129).
a(n) = A006012(n+1)/2 = A056236(n+1)/4. - Michael Somos, Mar 06 2003
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 3X3 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 by 2 matrix M=((1, 1), (1, 3)). - Simone Severini, Oct 15 2005
E.g.f. : exp(2x)(cosh(sqrt(2x)+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((-1)^k*binomial(2*n,n+4*k)/2, k=-floor(n/4)..floor(n/4)). [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
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EXAMPLE
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G.f. = 1 + 3*x + 10*x^2 + 34*x^3 + 116*x^4 + 396*x^5 + 1352*x^6 + 4616*x^7 + ...
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A006012, A003480, A056236.
First differences of A007070.
Cf. A000670, A001523, A060223, A227038, A328509, A332577, A332743, A332873.
Sequence in context: A255813 A113300 A332872 * A048580 A291292 A289612
Adjacent sequences: A007049 A007050 A007051 * A007053 A007054 A007055
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KEYWORD
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nonn,easy
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AUTHOR
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Colin Mallows, N. J. A. Sloane, Simon Plouffe
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STATUS
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approved
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