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A068156 G.f.: (x+2)*(x+1)/((x-1)*(x-2)) = Sum_{n>=0} a(n)*(x/2)^n. 27

%I

%S 1,3,9,21,45,93,189,381,765,1533,3069,6141,12285,24573,49149,98301,

%T 196605,393213,786429,1572861,3145725,6291453,12582909,25165821,

%U 50331645,100663293,201326589,402653181,805306365,1610612733

%N G.f.: (x+2)*(x+1)/((x-1)*(x-2)) = Sum_{n>=0} a(n)*(x/2)^n.

%C Number of moves to solve Hard Pagoda puzzle.

%C Partial sums of A111286. Binomial transform of (1,2,4,2,4,2,4 ....). - _Paul Barry_, Feb 28 2003

%C Warren W. Kokko writes that this sequence also appears to give the number of scoring sequences for the Racer Dice Game with n dice. - _N. J. A. Sloane_, Feb 24 2015

%C From _Michel Lagneau_, Apr 27 2015: (Start)

%C For n > 0, a(n) is the number of identical bowls having the same weight except for one which has a higher weight than the others which are identifiable by a weighing machine using n weighings.

%C Example: a(2)=9 because two weighings are sufficient:

%C Start with 9 bowls;

%C Step 1: remove 3 bowls => there are still 6 bowls;

%C Step 2: first weighing of 6 bowls (3 bowls on each side of the weighing machine);

%C Step 3: if the machine is in equilibrium, we find immediately the unknown bowl with a second weighing from the first 3 removing bowls. Else, we find immediately the unknown bowl with a second weighing from the 3 heaviest bowls.

%C Note: If the unknown bowl has a lower weight, the reasoning is the same, but it is necessary to know whether the unknown bowl is heavier or lighter.

%C In the general case, we always remove 3 bowls in step 1.

%C (End)

%C The number of ternary words of length n that avoid {11-2,22-1}. G.f. [1+(k-1)*x^2]/[1-k*x+(k-1)*x^2] at k=3. [Theorem 7.93 by Heubach and Mansour]. - _R. J. Mathar_, May 22 2016

%C Apart from the first term, column 2 of A222057. - _Anton Zakharov_, Oct 27 2016

%D Richard I. Hess, Compendium of Over 7000 Wire Puzzles, privately printed, 1991.

%D Richard I. Hess, Analysis of Ring Puzzles, booklet distributed at 13th International Puzzle Party, Amsterdam, Aug 20 1993.

%D S. Heubach, T. Mansour, in Combinatorics of Compositions and words, Discr. Math. Applicat. (ed by K H Rosen), CRC Press 2010, p 300.

%D Warren W. Kokko, The Racer Dice Game, Manuscript, 2015.

%H Vincenzo Librandi, <a href="/A068156/b068156.txt">Table of n, a(n) for n = 0..1000</a>

%H Artur Schaefer, <a href="http://arxiv.org/abs/1602.02186">Endomorphisms of The Hamming Graph and Related Graphs</a>, arXiv preprint arXiv:1602.02186 [math.CO], 2016. See Table Remark 4.5.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2).

%F a(0) = 1, a(n) = A060482(2n+1). For n > 0, a(n+1) = 2*a(n)+3.

%F G.f.: (1+2*x^2)/((1-2*x)*(1-x)). - _Paul Barry_, Feb 28 2003

%F a(n) = 3*2^n+0^n-3. - _Paul Barry_, Sep 04 2003

%F a(n) = A099257(A033484(n)+1) = 2*A033484(n) + 1. - _Reinhard Zumkeller_, Oct 09 2004

%F a(n) = 3*a(n-1) - 2*a(n-2), n > 1. - _Vincenzo Librandi_, Nov 11 2011

%F a(n) = a(n-1)+ 3*2^(n-1); a(1)=3. - _Ctibor O. Zizka_, Apr 17 2008

%F E.g.f.: 1 + 3*(exp(x) - 1)*exp(x). - _Ilya Gutkovskiy_, May 22 2016

%t Join[{1}, LinearRecurrence[{3, -2}, {3, 9}, 30]] (* _Jean-Fran├žois Alcover_, Jan 08 2019 *)

%o (MAGMA) [3*2^n+0^n-3 : n in [0..30]]; // _Vincenzo Librandi_, Nov 11 2011

%Y A diagonal of A233308 (for n > 1).

%K nonn,easy

%O 0,2

%A _Benoit Cloitre_, Mar 12 2002

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Last modified July 31 06:15 EDT 2021. Contains 346369 sequences. (Running on oeis4.)