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A068293 a(1) = 1; thereafter a(n) = 6*(2^(n-1) - 1).

%I A068293 #49 Oct 19 2024 15:57:32
%S A068293 1,6,18,42,90,186,378,762,1530,3066,6138,12282,24570,49146,98298,
%T A068293 196602,393210,786426,1572858,3145722,6291450,12582906,25165818,
%U A068293 50331642,100663290,201326586,402653178,805306362,1610612730,3221225466,6442450938,12884901882
%N A068293 a(1) = 1; thereafter a(n) = 6*(2^(n-1) - 1).
%C A068293 1/4 the number of colorings of an n X n octagonal array with 4 colors.
%C A068293 Consider the planar net 3^6 (as in the top left figure in the uniform planar nets link). Then a(n) is the total number of ways that a spider starting at a point P can reach any point n steps away by using a path of length n. - _N. J. A. Sloane_, Feb 20 2016
%C A068293 From _Gary W. Adamson_, Jan 13 2009: (Start)
%C A068293 Equals inverse binomial transform of A091344: (1, 7, 31, 115, 391, ...).
%C A068293 Equals binomial transform of (1, 5, 7, 5, 7, 5, ...). (End)
%C A068293 For n > 1, number of ternary strings of length n with exactly 2 different digits. - _Enrique Navarrete_, Nov 20 2020
%H A068293 Harvey P. Dale, <a href="/A068293/b068293.txt">Table of n, a(n) for n = 1..1000</a>
%H A068293 Ana Rechtman, <a href="http://images-archive.math.cnrs.fr/Fevrier-2016-3e-defi.html">Février 2016, 3e défi</a>, Images des Mathématiques, CNRS, 2016.
%H A068293 N. J. A. Sloane, <a href="/A008576/a008576.png">The uniform planar nets and their A-numbers</a> [Annotated scanned figure from Gruenbaum and Shephard (1977)]
%H A068293 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2).
%F A068293 G.f.: (1+x)*(1+2*x)/((1-x)*(1-2*x)). - _Benoit Cloitre_, Apr 13 2002
%F A068293 a(n) = 3*a(n-1) - 2*a(n-2); a(1)=1, a(2)=6, a(3)=18. - _Harvey P. Dale_, Nov 27 2011
%F A068293 E.g.f.: 1 - 6*exp(x)*(exp(x) - 1). - _Stefano Spezia_, May 18 2024
%t A068293 a=0; lst={1}; k=6; Do[a+=k; AppendTo[lst, a]; k+=k, {n, 0, 5!}]; lst (* _Vladimir Joseph Stephan Orlovsky_, Dec 16 2008 *)
%t A068293 Transpose[NestList[{First[#]+1,6(2^First[#]-1)}&,{1,1},30]][[2]] (* or *) Join[{1},LinearRecurrence[{3,-2},{6,18},30]] (* _Harvey P. Dale_, Nov 27 2011 *)
%o A068293 (PARI) a(n)=polcoeff(prod(i=1,2,(1+i*x))/(prod(i=1,2,(1-i*x))+x*O(x^n)),n)
%o A068293 for(n=0,50,print1(a(n),","))
%o A068293 (Magma) [1] cat [6*(2^(n-1)-1): n in [2..40]]; // _Vincenzo Librandi_, Feb 20 2016
%Y A068293 Cf. A068239-A068305, A000332, A002417, A027441.
%Y A068293 Cf. A091344. - _Gary W. Adamson_, Jan 13 2009
%K A068293 nonn,easy
%O A068293 1,2
%A A068293 _R. H. Hardin_, Feb 24 2002
%E A068293 More terms from _Benoit Cloitre_, Apr 13 2002
%E A068293 Old definition (which is now a comment) replaced with explicit formula by _N. J. A. Sloane_, May 12 2010