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

%I A068156 #96 Jan 11 2025 04:47:29
%S A068156 1,3,9,21,45,93,189,381,765,1533,3069,6141,12285,24573,49149,98301,
%T A068156 196605,393213,786429,1572861,3145725,6291453,12582909,25165821,
%U A068156 50331645,100663293,201326589,402653181,805306365,1610612733
%N A068156 G.f.: (x+2)*(x+1)/((x-1)*(x-2)) = Sum_{n>=0} a(n)*(x/2)^n.
%C A068156 Number of moves to solve Hard Pagoda puzzle.
%C A068156 Partial sums of A111286. Binomial transform of (1,2,4,2,4,2,4 ....). - _Paul Barry_, Feb 28 2003
%C A068156 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 A068156 From _Michel Lagneau_, Apr 27 2015: (Start)
%C A068156 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 A068156 Example: a(2)=9 because two weighings are sufficient:
%C A068156 Start with 9 bowls;
%C A068156 Step 1: remove 3 bowls => there are still 6 bowls;
%C A068156 Step 2: first weighing of 6 bowls (3 bowls on each side of the weighing machine);
%C A068156 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 A068156 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 A068156 In the general case, we always remove 3 bowls in step 1.
%C A068156 (End)
%C A068156 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 A068156 Apart from the first term, column 2 of A222057. - _Anton Zakharov_, Oct 27 2016
%D A068156 Richard I. Hess, Compendium of Over 7000 Wire Puzzles, privately printed, 1991.
%D A068156 Richard I. Hess, Analysis of Ring Puzzles, booklet distributed at 13th International Puzzle Party, Amsterdam, Aug 20 1993.
%D A068156 S. Heubach, T. Mansour, in Combinatorics of Compositions and words, Discr. Math. Applicat. (ed by K H Rosen), CRC Press 2010, p 300.
%D A068156 Warren W. Kokko, The Racer Dice Game, Manuscript, 2015.
%H A068156 Vincenzo Librandi, <a href="/A068156/b068156.txt">Table of n, a(n) for n = 0..1000</a>
%H A068156 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 A068156 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2).
%F A068156 a(0) = 1, a(n) = A060482(2n+1). For n > 0, a(n+1) = 2*a(n)+3.
%F A068156 G.f.: (1+2*x^2)/((1-2*x)*(1-x)). - _Paul Barry_, Feb 28 2003
%F A068156 a(n) = 3*2^n+0^n-3. - _Paul Barry_, Sep 04 2003
%F A068156 a(n) = A099257(A033484(n)+1) = 2*A033484(n) + 1. - _Reinhard Zumkeller_, Oct 09 2004
%F A068156 a(n) = 3*a(n-1) - 2*a(n-2), n > 1. - _Vincenzo Librandi_, Nov 11 2011
%F A068156 a(n) = a(n-1)+ 3*2^(n-1); a(1)=3. - _Ctibor O. Zizka_, Apr 17 2008
%F A068156 E.g.f.: 1 + 3*(exp(x) - 1)*exp(x). - _Ilya Gutkovskiy_, May 22 2016
%t A068156 Join[{1}, LinearRecurrence[{3, -2}, {3, 9}, 30]] (* _Jean-François Alcover_, Jan 08 2019 *)
%t A068156 CoefficientList[Series[(1+2x^2)/((1-2x)(1-x)),{x,0,40}],x] (* _Harvey P. Dale_, Jan 02 2022 *)
%o A068156 (Magma) [3*2^n+0^n-3 : n in [0..30]]; // _Vincenzo Librandi_, Nov 11 2011
%o A068156 (Sage) def a(n): return 3*2**n+0**n-3 # _Torlach Rush_, Jan 09 2025
%Y A068156 A diagonal of A233308 (for n > 1).
%Y A068156 Cf. A000079.
%K A068156 nonn,easy
%O A068156 0,2
%A A068156 _Benoit Cloitre_, Mar 12 2002