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A057711
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a(0)=0, a(1)=1, a(n)=n*2^(n-2) for n>=2.
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31
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Number of states in the planning domain FERRY, when n-3 cars are at one of two shores while the (n-2)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)= 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*A119458(n,k),k=0..n). - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 20 2006
a(n) = 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 (callan(AT)stat.wisc.edu), Mar 30 2007
a(n) is the total number of runs in all Boolean (n-1)-strings. For example, the 8 Boolean 3-strings, 000, 001, 010, 011, 100, 101, 110, 111 have 1, 2, 3, 2, 2, 3, 2, 1 runs respectively. - David Callan (callan(AT)stat.wisc.edu), Jul 22 2008
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), 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)
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Aug 15 2010: (Start)
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).
(End)
Starting with 1 = (1, 1, 2, 4, 8, 16,...) convolved with (1, 1, 3, 7, 15, 31,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 26 2010]
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REFERENCES
| M. Ghallab, A. Howe et al., PDDL - The Planning Domain Definition Language, Version 1.2. Technical Report CVC TR-98-003/DCS TR-1165. Yale Center for Computational Vision and Control, 1998
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..1000
P. Chinn, R. Grimaldi and S. Heubach, The frequency of summands of a particular size ..., Ars Combin. 69 (2003), 65-78.
M. Ghallab et al., FERRY domain
B. Wolf, Creating state sets
Index to sequences with linear recurrences with constant coefficients, signature (4,-4).
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FORMULA
| a(n) = ceiling(n*2^(n-2)).
Binomial transform of (0, 1, 0, 3, 0, 5, 0, 7, ...)
a(0)=0, a(n)=n(0^(n-1)+2^(n-1))/2, n>0 a(n)=sum{k=0..n, C(n, 2k+1)(2k+1) }. E.g.f. xexp(x)cosh(x) (starts 0, 1, 6, ...). - Paul Barry (pbarry(AT)wit.ie), Apr 06 2003
The sequence 1, 1, 6, 16, ... is the binomial transform of A016813 with interpolated zeros. - Paul Barry (pbarry(AT)wit.ie), Jul 25 2003
For n>1, a(n)=Sum((k-n/2)^2 C(n, k), k=0, .., n). (n+1)a(n)=A001788(n). - Mario Catalani (mario.catalani(AT)unito.it), Nov 26 2003
a(n)=n2^(n-2)-sum{k=0..n, binom(n, k)k*(-1)^k}. G.f.: x(1-2x+2x^2)/(1-2x)^2. - Paul Barry (pbarry(AT)wit.ie), May 07 2004
ceil(binomial(n+1,1)*2^(n-1)). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 01 2006
a(n+1)=Sum_{0<=k<=n} A196389(n,k)*2^k. - From DELEHAM Philippe, Oct 31 2011
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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
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MAPLE
| [seq (ceil(binomial(n+1, 1)*2^(n-1)), n=-1..29)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 01 2006
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MATHEMATICA
| Table[(n+2) 2^n, {n, 0, 30}]
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PROG
| (MAGMA) [Ceiling(n*2^(n-2)) : n in [0..40]]; // Vincenzo Librandi, Sep 22 2011
(PARI) a(n)=ceil(n*2^(n-2)) \\ Charles R Greathouse IV, Oct 31 2011
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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, A057711.
Sequence in context: A078774 A174016 A129952 * A111281 A018021 A074405
Adjacent sequences: A057708 A057709 A057710 * A057712 A057713 A057714
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KEYWORD
| easy,nonn
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AUTHOR
| Bernhard Wolf (wolf(AT)cs.tu-berlin.de), Oct 24 2000
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Oct 25 2000
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