%I #48 May 18 2024 14:53:30

%S 1,6,18,42,90,186,378,762,1530,3066,6138,12282,24570,49146,98298,

%T 196602,393210,786426,1572858,3145722,6291450,12582906,25165818,

%U 50331642,100663290,201326586,402653178,805306362,1610612730,3221225466,6442450938,12884901882

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

%C 1/4 the number of colorings of an n X n octagonal array with 4 colors.

%C 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 From _Gary W. Adamson_, Jan 13 2009: (Start)

%C Equals inverse binomial transform of A091344: (1, 7, 31, 115, 391, ...).

%C Equals binomial transform of (1, 5, 7, 5, 7, 5, ...). (End)

%C For n > 1, number of ternary strings of length n with exactly 2 different digits. - _Enrique Navarrete_, Nov 20 2020

%H Harvey P. Dale, <a href="/A068293/b068293.txt">Table of n, a(n) for n = 1..1000</a>

%H Ana Rechtman, <a href="http://images.math.cnrs.fr/Fevrier-2016-3e-defi.html">Février 2016, 3e défi</a>, Images des Mathématiques, CNRS, 2016.

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

%F G.f.: (1+x)*(1+2*x)/((1-x)*(1-2*x)). - _Benoit Cloitre_, Apr 13 2002

%F 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 E.g.f.: 1 - 6*exp(x)*(exp(x) - 1). - _Stefano Spezia_, May 18 2024

%t 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 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 (PARI) a(n)=polcoeff(prod(i=1,2,(1+i*x))/(prod(i=1,2,(1-i*x))+x*O(x^n)),n)

%o for(n=0,50,print1(a(n),","))

%o (Magma) [1] cat [6*(2^(n-1)-1): n in [2..40]]; // _Vincenzo Librandi_, Feb 20 2016

%Y Cf. A068239-A068305, A000332, A002417, A027441.

%Y Cf. A091344. - _Gary W. Adamson_, Jan 13 2009

%K nonn,easy

%O 1,2

%A _R. H. Hardin_, Feb 24 2002

%E More terms from _Benoit Cloitre_, Apr 13 2002

%E Old definition (which is now a comment) replaced with explicit formula by _N. J. A. Sloane_, May 12 2010