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A007052
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Number of order-consecutive partitions of n.
(Formerly M2847)
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22
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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 (layman(AT)math.vt.edu), 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 (kociemba(AT)t-online.de), Jun 12 2004
Equals the INVERT transform of (1, 2, 5, 13, 34, 89,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 01 2009]
a(n) is the number of compositions of n when there are 3 types of ones. [From Milan R. Janjic (agnus(AT)blic.net), Aug 13 2010]
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REFERENCES
| Hwang, F. K.; Mallows, C. L.; Enumerating nested and consecutive partitions. J. Combin. Theory Ser. A 70 (1995), no. 2, 323-333.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 164
N. J. A. Sloane, Transforms
Index entries for sequences related to poker
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FORMULA
| 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).
G.f.: (1-x)/(1-4x+2x^2).
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 (pbarry(AT)wit.ie), 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 (pbarry(AT)wit.ie), Nov 22 2003
a(n)=( (2-Sqrt(2))^(n+1)+(2+Sqrt(2))^(n+1) )/4. - Herbert Kociemba (Kociemba(AT)t-online.de), 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 (qntmpkt(AT)yahoo.com), 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 (simoseve(AT)gmail.com), Oct 15 2005
E.g.f. : exp(2x)(cosh(sqrt(2x)+sinh(sqrt(2)x)/sqrt(2) - Paul Barry (pbarry(AT)wit.ie), Nov 20 2003
a(n)=A007068(2*n), n>0. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 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. [From Milan R. Janjic (agnus(AT)blic.net), 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]
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MATHEMATICA
| a[n_]:=(MatrixPower[{{3, 1}, {1, 1}}, n].{{2}, {1}})[[2, 1]]; Table[a[n], {n, 0, 40}] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Feb 20 2010]
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PROG
| (PARI) a(n)=real((2+quadgen(8))^(n+1))/2
(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
| a(n)=A006012(n+1)/2=A056236(n+1)/4. Cf. A003480.
Cf. A007070.
Sequence in context: A193036 A083580 A113300 * A048580 A059738 A094832
Adjacent sequences: A007049 A007050 A007051 * A007053 A007054 A007055
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KEYWORD
| nonn,easy
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AUTHOR
| Colin Mallows, N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
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