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A006043 A traffic light problem: expansion of 2/(1 - 3*x)^3.

%I A006043 M2107 #71 Jan 08 2023 02:39:40
%S A006043 2,18,108,540,2430,10206,40824,157464,590490,2165130,7794468,27634932,
%T A006043 96722262,334807830,1147912560,3902902704,13172296626,44165935746,
%U A006043 147219785820,488149816140,1610894393262,5292938720718,17322344904168,56485907296200,183579198712650,594796603828986
%N A006043 A traffic light problem: expansion of 2/(1 - 3*x)^3.
%C A006043 Column 2 of square array A152818. - _Omar E. Pol_, Jan 05 2009
%C A006043 In [Bach et al., Section 9], 2*a(n-2) counts the "small diagrams". - _Eric M. Schmidt_, Sep 23 2017
%D A006043 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006043 Vincenzo Librandi, <a href="/A006043/b006043.txt">Table of n, a(n) for n = 0..1000</a>
%H A006043 Eric Bach, Jeremie Dusart, Lisa Hellerstein, and Devorah Kletenik, <a href="https://arxiv.org/abs/1702.04067">Submodular Goal Value of Boolean Functions</a>, arXiv:1702.04067 [cs.DM], 2017.
%H A006043 Frank A. Haight, <a href="http://www.jstor.org/stable/2333538">Overflow at a traffic light</a>, Biometrika, 46 (1959), 420-424.
%H A006043 Frank A. Haight, <a href="/A001787/a001787_3.pdf">Overflow at a traffic light</a>, Biometrika, 46 (1959), 420-424. (Annotated scanned copy)
%H A006043 Frank A. Haight, <a href="/A001787/a001787_2.pdf">Letter to N. J. A. Sloane, n.d.</a>.
%H A006043 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (9,-27,27).
%F A006043 a(n) = (n+2)*(n+1)*3^n. - _Zerinvary Lajos_, Apr 25 2007, corrected by _R. J. Mathar_, Mar 14 2011
%F A006043 a(n) = 2*A027472(n+3) = A116138(n+1)/3. - _R. J. Mathar_, Mar 14 2011
%F A006043 a(n) = 2*A000217(n+1)*A000244(n). - _Zak Seidov_, Mar 14 2011
%F A006043 E.g.f.: exp(3*x)*(2 + 12*x + 9*x^2). - _Stefano Spezia_, Jan 01 2023
%F A006043 From _Amiram Eldar_, Jan 08 2023: (Start)
%F A006043 Sum_{n>=0} 1/a(n) = 3 - 6*log(3/2).
%F A006043 Sum_{n>=0} (-1)^n/a(n) = 12*log(4/3) - 3. (End)
%p A006043 seq((n+2)*(n+1)*3^n, n=0..23); # _Zerinvary Lajos_, Apr 25 2007
%t A006043 f[n_] := (n + 2) (n + 1) 3^n; Array[f, 22, 0] (* _Robert G. Wilson v_, Mar 15 2011 *)
%t A006043 CoefficientList[Series[2/(1 - 3 x)^3, {x, 0, 21}], x] (* _Robert G. Wilson v_, Mar 15 2011 *)
%t A006043 LinearRecurrence[{9,-27,27},{2,18,108},30] (* _Harvey P. Dale_, Apr 27 2017 *)
%o A006043 (PARI) a(n)=(n+2)*(n+1)*3^n \\ _Charles R Greathouse IV_, Mar 16 2011
%o A006043 (Magma)[(n+2)*(n+1)*3^n: n in [0..30]]; // _Vincenzo Librandi_, Aug 15 2011
%Y A006043 Cf. A006044, A000142, A152818, A154120. - _Omar E. Pol_, Jan 05 2009
%Y A006043 Cf. A000217, A000244, A027472, A116138,
%K A006043 nonn,easy
%O A006043 0,1
%A A006043 _N. J. A. Sloane_, _Simon Plouffe_