

A057711


a(0)=0, a(1)=1, a(n) = n*2^(n2) for n >= 2.


44



0, 1, 2, 6, 16, 40, 96, 224, 512, 1152, 2560, 5632, 12288, 26624, 57344, 122880, 262144, 557056, 1179648, 2490368, 5242880, 11010048, 23068672, 48234496, 100663296, 209715200, 436207616, 905969664, 1879048192, 3892314112, 8053063680
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OFFSET

0,3


COMMENTS

Number of states in the planning domain FERRY, when n3 cars are at one of two shores while the (n2)nd car may be on the ferry or at one of the shores.
If the ferry could board any number of cars (instead of only one), the number of states would form the Pisot sequence P(2,6) (A008776). In addition, if k shores existed, the sequence would form the Pisot sequence P(k,k(k+1)). This corresponds to the BRIEFCASE planning domain.
a(i) is the number of occurrences of the number 1 in all palindromic compositions of n = 2*(i+1).  Silvia Heubach (sheubac(AT)calstatela.edu), Jan 10 2003. E.g., there are 5 palindromic compositions of 6, namely 111111 11211 2112 1221 141, containing a total of 16 1's.
Number of occurrences of 00's in all circular binary words of length n. Example: a(3)=6 because in the circular binary words 000, 001, 010, 011, 100, 101, 110 and 111 we have a total of 3+1+1+0+1+0+0+0=6 occurrences of 00. a(n) = Sum_{k=0..n} k*A119458(n,k).  Emeric Deutsch, May 20 2006
a(n) is the number of permutations on [n] for which the entries of each left factor form a circular subinterval of [n]. A subset I of [n] forms a circular subinterval of [n] if it is an ordinary interval [a,b] or has the form [1,a]union[b,n] for 1 <= a < b <= n. For example, (5,4,2) is a left factor of the permutation (5,4,2,1,3) which does not form a circular subinterval of [5] and a(4)=16 counts all 24 permutations of [4] except the eight whose first two entries are 1,3 (in either order) or 2,4.  David Callan, Mar 30 2007
a(n) is the total number of runs in all Boolean (n1)strings. For example, the 8 Boolean 3strings, 000, 001, 010, 011, 100, 101, 110, 111 have 1, 2, 3, 2, 2, 3, 2, 1 runs respectively.  David Callan, Jul 22 2008
From Gary W. Adamson, Jul 31 2010: (Start)
Starting with "1" = (1, 2, 4, 8, ...) convolved with (1, 0, 2, 4, 8, ...).
Example: a(6) = 96 = (32, 16, 8, 4, 2, 1) dot (1, 0, 2, 4, 8, 16) = (32 + 0 + 16 + 16 + 16, + 16) = 32 + 4*16 (End)
An elephant sequence, see A175654. For the corner squares 24 A[5] vectors, with decimal values between 27 and 432, lead to this sequence (without the leading 0). For the central square these vectors lead to the companion sequence A087447 (without the first leading 1).  Johannes W. Meijer, Aug 15 2010
Starting with 1 = (1, 1, 2, 4, 8, 16, ...) convolved with (1, 1, 3, 7, 15, 31, ...).  Gary W. Adamson, Oct 26 2010
a(n) is the number of ways to draw simple polygonal chains for n vertices lying on a circle.  Anton Zakharov, Dec 31 2016
Also the number of edges, maximal cliques, and maximum cliques in the nfolded cube graph for n > 3.  Eric W. Weisstein, Dec 01 2017 and Mar 21 2018
Number of pairs of compositions of n corresponding to a seaweed algebra of index n2 for n > 2.  Nick Mayers, Jun 25 2018
Starting with 1, 2, 6, 16, ..., number of permutations of length n>0 avoiding the partially ordered pattern (POP) {1>2, 1>3} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the first element is larger than the second and third elements.  Sergey Kitaev, Dec 08 2020


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
O. Aichholzer, A. Asinowski, T. Miltzow, Disjoint compatibility graph of noncrossing matchings of points in convex position, arXiv preprint arXiv:1403.5546 [math.CO], 2014.
A. Burstein, S. Kitaev, T. Mansour, Partially ordered patterns and their combinatorial interpretations, PU. M. A. Vol. 19 (2008), No. 23, pp. 2738.
P. Chinn, R. Grimaldi and S. Heubach, The frequency of summands of a particular size in Palindromic Compositions, Ars Combin. 69 (2003), 6578.
Vincent Coll et al., Meander graphs and Frobenius seaweed Lie algebras II, Journal of Generalized Lie Theory and Applications 9.1 (2015).
Vladimir Dergachev, and Alexandre Kirillov, Index of Lie algebras of seaweed type, J. Lie Theory 10.2 (2000): 331343.
Alice L. L. Gao, Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao, Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
M. Ghallab et al., FERRY domain
M. Ghallab, A. Howe et al., PDDL  The Planning Domain Definition Language, Version 1.2, Technical Report CVC TR98003/DCS TR1165. Yale Center for Computational Vision and Control, 1998.
Eric Weisstein's World of Mathematics, Folded Cube Graph
Eric Weisstein's World of Mathematics, Maximal Clique
Eric Weisstein's World of Mathematics, Maximum Clique
B. Wolf, Creating state sets
Index entries for linear recurrences with constant coefficients, signature (4,4).


FORMULA

a(n) = ceiling(n*2^(n2)).
Binomial transform of (0, 1, 0, 3, 0, 5, 0, 7, ...).
From Paul Barry, Apr 06 2003: (Start)
a(0)=0, a(n) = n*(0^(n1) + 2^(n1))/2, n > 0.
a(n) = Sum_{k=0..n} binomial(n, 2k+1)*(2k+1).
E.g.f.: x*exp(x)*cosh(x). (End)
The sequence 1, 1, 6, 16, ... is the binomial transform of A016813 with interpolated zeros.  Paul Barry, Jul 25 2003
For n > 1, a(n) = Sum_{k=0..n} (kn/2)^2 C(n, k). (n+1)*a(n) = A001788(n).  Mario Catalani (mario.catalani(AT)unito.it), Nov 26 2003
From Paul Barry, May 07 2004: (Start)
a(n) = n*2^(n2)  Sum_{k=0..n} binomial(n, k)*k*(1)^k.
G.f.: x*(12*x+2*x^2)/(12*x)^2. (End)
a(n+1) = ceiling(binomial(n+1,1)*2^(n1)).  Zerinvary Lajos, Nov 01 2006
a(n+1) = Sum_{k=0..n} A196389(n,k)*2^k.  Philippe Deléham, Oct 31 2011
a(0)=0, a(1)=1, a(2)=2, a(3)=6, a(n+1) = 4*a(n)4*a(n1) for n >= 3.  Philippe Deléham, Feb 20 2013
a(n) = A002064(n1)  A002064(n2), for n >= 2.  Ivan N. Ianakiev, Dec 29 2013
From Amiram Eldar, Aug 05 2020: (Start)
Sum_{n>=1} 1/a(n) = 4*log(2)  1.
Sum_{n>=1} (1)^(n+1)/a(n) = 4*log(3/2)  1. (End)


EXAMPLE

a(1)=6 because the palindromic compositions of n=4 are 4, 1+2+1, 1+1+1+1 and 2+2 and they contain 6 ones.  Silvia Heubach (sheubac(AT)calstatela.edu), Jan 10 2003


MATHEMATICA

Join[{0, 1}, Table[n 2^(n  2), {n, 2, 30}]] (* Eric W. Weisstein, Dec 01 2017 *)
Join[{0, 1}, LinearRecurrence[{4, 4}, {2, 6}, 20]] (* Eric W. Weisstein, Dec 01 2017 *)
CoefficientList[Series[x (1  2 x + 2 x^2)/(1  2 x)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Dec 01 2017 *)


PROG

(MAGMA) [Ceiling(n*2^(n2)) : n in [0..40]]; // Vincenzo Librandi, Sep 22 2011
(PARI) a(n)=ceil(n*2^(n2)) \\ Charles R Greathouse IV, Oct 31 2011
(PARI) x='x+O('x^50); concat(0, Vec(x*(12*x+2*x^2)/(12*x)^2)) \\ Altug Alkan, Nov 01 2015


CROSSREFS

Cf. A082133, A082134, A082135, A082136.
Pisot sequence P(2, 6) (A008776), Pisot sequence P(k, k(k+1))
Cf. A119458.
Cf. A082140, A082141, A082138, A082139, A080951, A080929.
Sequence in context: A174016 A265725 A129952 * A302239 A264551 A293004
Adjacent sequences: A057708 A057709 A057710 * A057712 A057713 A057714


KEYWORD

easy,nonn


AUTHOR

Bernhard Wolf (wolf(AT)cs.tuberlin.de), Oct 24 2000


STATUS

approved



