Search: id:A057711
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%I A057711
%S A057711 0,1,2,6,16,40,96,224,512,1152,2560,5632,12288,26624,57344,122880,262144,
%T A057711 557056,1179648,2490368,5242880,11010048,23068672,48234496,100663296,
%U A057711 209715200,436207616,905969664,1879048192,3892314112,8053063680
%N A057711 a(0)=0, a(1)=1, a(n)=n*2^(n-2) for n>=2.
%C A057711 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.
%C A057711 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.
%C A057711 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.
%C A057711 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
%C A057711 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
%C A057711 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
%D A057711 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
%H A057711 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A057711 P. Chinn, R. Grimaldi and S. Heubach, <a href="http://www.calstatela.edu/
               faculty/sheubac/papers/Freqs.pdf">The frequency of summands of a 
               particular size ...</a>, Ars Combin. 69 (2003), 65-78.
%H A057711 M. Ghallab et al., <a href="ftp://ftp.cs.yale.edu/pub/mcdermott/software/
               pddl.tar.gz">FERRY domain</a>
%H A057711 B. Wolf, <a href="http://ki.cs.tu-berlin.de/diplomarbeiten/wolf/da.html">
               Creating state sets</a>
%F A057711 a(n) = ceiling(n*2^(n-2)).
%F A057711 Binomial transform of (0, 1, 0, 3, 0, 5, 0, 7, ...)
%F A057711 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
%F A057711 The sequence 1, 1, 6, 16, ... is the binomial transform of A016813 with 
               interpolated zeros. - Paul Barry (pbarry(AT)wit.ie), Jul 25 2003
%F A057711 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
%F A057711 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
%F A057711 ceil(binomial(n+1,1)*2^(n-1)). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Nov 01 2006
%e A057711 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
%p A057711 [seq (ceil(binomial(n+1,1)*2^(n-1)),n=-1..29)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Nov 01 2006
%t A057711 Table[(n+2) 2^n, {n, 0, 30}]
%Y A057711 Cf. A082133, A082134, A082135, A082136.
%Y A057711 Pisot sequence P(2, 6) (A008776), Pisot sequence P(k, k(k+1))
%Y A057711 Cf. A119458.
%Y A057711 Cf. A082140, A082141, A082138, A082139, A080951, A080929, A057711.
%Y A057711 Sequence in context: A046209 A078774 A129952 this_sequence A111281 A018021 
               A074405
%Y A057711 Adjacent sequences: A057708 A057709 A057710 this_sequence A057712 A057713 
               A057714
%K A057711 easy,nonn
%O A057711 0,3
%A A057711 Bernhard Wolf (wolf(AT)cs.tu-berlin.de), Oct 24 2000
%E A057711 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Oct 25 2000

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