%I #29 Sep 08 2022 08:46:06
%S 16,10,36,82,272,890,3108,11042,39952,146026,537636,1988722,7379216,
%T 27436250,102144036,380604482,1418981392,5292200650,19742287908,
%U 73658763922,274848860432,1025630676026,3827417932836,14283423231842,53304783436816,198932109576490,742414961433636,2770706748348082
%N a(n) = 6 + 4*(-1)^n + (2+sqrt(3))^n + (2-sqrt(3))^n + 2*(1+sqrt(2))^n + 2*(1-sqrt(2))^n.
%H G. C. Greubel, <a href="/A232999/b232999.txt">Table of n, a(n) for n = 0..1000</a>
%H D. Deford, <a href="http://www.math.dartmouth.edu/~ddeford/graphs.pdf">Seating rearrangements on arbitrary graphs</a>, preprint 2013, Involve, Volume 7, Number 6 (2014), 787-805. See Table 2.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-7,-8,9,2,-1).
%F G.f.: -2*(x^5+16*x^4-32*x^3-44*x^2+43*x-8) / ((x-1)*(x+1)*(x^2-4*x+1)*(x^2+2*x-1)). - _Colin Barker_, Sep 12 2014
%F a(n) = 6*a(n-1) - 7*a(n-2) - 8*a(n-3) + 9*a(n-4) + 2*a(n-5) - a(n-6) for n >= 6. - _Nathaniel Johnston_, Sep 12 2014
%F E.g.f.: 4*exp(-x) + 6*exp(x) + 2*exp((1 + sqrt(2))*x) + exp(-(-2 + sqrt(3))*x) + exp((2 + sqrt(3))*x) + 2*exp(x - sqrt(2)*x). - _Stefano Spezia_, Oct 07 2018
%p f:=n->expand(6+4*(-1)^n
%p +(2+sqrt(3))^n
%p +(2-sqrt(3))^n
%p +2*(1+sqrt(2))^n
%p +2*(1-sqrt(2))^n);
%p [seq(f(n),n=0..30)];
%t LinearRecurrence[{6,-7,-8,9,2,-1},{16,10,36,82,272,890},30] (* _Harvey P. Dale_, Mar 10 2015 *)
%t Simplify[CoefficientList[Series[4 E^-x + 6 E^x + 2 E^((1 + Sqrt[2]) x) + E^(-(-2 + Sqrt[3]) x) + E^((2 + Sqrt[3]) x) + 2 E^(x - Sqrt[2] x), {x, 0, 20}], x]*Table[k!, {k, 0, 30}]] (* _Stefano Spezia_, Oct 07 2018 *)
%o (PARI) Vec(-2*(x^5+16*x^4-32*x^3-44*x^2+43*x-8)/((x-1)*(x+1)*(x^2-4*x+1)*(x^2+2*x-1)) + O(x^100)) \\ _Colin Barker_, Sep 12 2014
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(-2*(x^5+16*x^4-32*x^3-44*x^2+43*x-8)/((x-1)*(x+1)*(x^2-4*x+1)*(x^2+2*x-1)))); // _G. C. Greubel_, Oct 06 2018
%o (GAP) a:=[16,10,36,82,272,890];; for n in [7..30] do a[n]:=6*a[n-1]-7*a[n-2]-8*a[n-3]+9*a[n-4]+2*a[n-5]-a[n-6]; od; a; # _Muniru A Asiru_, Oct 07 2018
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_, Dec 17 2013