%I #57 Dec 02 2023 15:23:13
%S 1,3,11,35,111,343,1051,3195,9671,29183,87891,264355,794431,2386023,
%T 7163531,21501515,64526391,193622863,580955971,1743042675,5229477551,
%U 15689131703,47068793211,141209175835,423633119911,1270910544543,3812754003251,11438306748995
%N Expansion of (1-x)/((1+x)*(1-2*x)*(1-3*x)).
%C Number of paths with n+2 steps on the cycle graph C_6 which start at the first node and end at the 3rd node and each step is -1, 0 or +1. - _Herbert Kociemba_, Sep 30 2020
%H Vincenzo Librandi, <a href="/A004054/b004054.txt">Table of n, a(n) for n = 0..1000</a>
%H Xavier Acloque, <a href="https://web.archive.org/web/20040920090251/http://members.fortunecity.fr/polynexus/index.html">Polynexus Numbers and other mathematical wonders</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-1,-6).
%F From _Paul Barry_, Sep 13 2003: (Start)
%F The sequence 0, 0, 1, ... has a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k)*A001045(2*k).
%F a(n) = 3^n/6 + (-1)^n/6 - 0^n/6 - 2^n/6. (End)
%F The signed sequence 0, 1, -3, ... has g.f. x*(1+x)/((1-x)*(1+2*x)*(1+3*x)) and a(n) = 1/6 + (-2)^n/3 - (-3)^n/2. It is the third inverse binomial transform of A001045(2*n-1) - 0^n/2. - _Paul Barry_, Apr 21 2004
%F From _Paul Barry_, Jul 22 2004: (Start)
%F Convolution of A000244 and A078008.
%F a(n) = Sum_{k=0..n} A078008(k)*3^(n-k).
%F a(n) = (3*A000244(n) - A001045(n+2))/2. (End)
%F a(n) = (A001047(n+2) + (-1)^n)/6. - _Vladimir Pletser_, Dec 02 2023
%F a(n) = A094705(n+1)-A094705(n). - _R. J. Mathar_, Dec 02 2023
%t Table[1/6 ((-1)^(2+n)-2^(n+2)+3^(n+2)),{n,0,30}] (* _Herbert Kociemba_, Sep 30 2020 *)
%o (Magma) [Ceiling(3^(n+2)/6+(-1)^(n+2)/6-0^n/6-2^(n+2)/6) : n in [0..30]]; // _Vincenzo Librandi_, Oct 08 2011
%o (PARI) Vec((1-x)/((1+x)*(1-2*x)*(1-3*x))+O(x^99)) \\ _Charles R Greathouse IV_, Sep 26 2012
%Y Cf. A001045, A001047.
%Y Cf. A000244, A078008.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_