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Expansion of 1/(1 - x - x^2 - 3*x^4 + 4*x^5 - 2*x^6).
1

%I #18 Sep 08 2022 08:45:51

%S 1,1,2,3,8,10,22,35,73,112,227,376,726,1216,2321,3981,7430,12907,

%T 23888,41886,76782,135631,247309,438860,796747,1419144,2568858,

%U 4586608,8284885,14819657,26728034,47870371,86244344,154607362,278326950,499272603,898307169

%N Expansion of 1/(1 - x - x^2 - 3*x^4 + 4*x^5 - 2*x^6).

%C The ratio a(n+1)/a(n) approaches 1.796757012458598901977511048324681177...

%H G. C. Greubel, <a href="/A175715/b175715.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,3,-4,2).

%F G.f.: 1/(1 - x - x^2 - 3*x^4 + 4*x^5 - 2*x^6).

%p seq(coeff(series(1/(1-x-x^2-3*x^4+4*x^5-2*x^6), x, n+1), x, n), n = 0..40); # _G. C. Greubel_, Dec 04 2019

%t LinearRecurrence[{1,1,0,3,-4,2}, {1,1,2,3,8,10}, 40] (* _Bruno Berselli_, May 17 2017 *)

%o (PARI) my(x='x+O('x^40)); Vec(1/(1-x-x^2-3*x^4+4*x^5-2*x^6)) \\ _G. C. Greubel_, Dec 04 2019

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1-x-x^2-3*x^4+4*x^5-2*x^6) )); // _G. C. Greubel_, Dec 04 2019

%o (Sage)

%o def A175715_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P( 1/(1-x-x^2-3*x^4+4*x^5-2*x^6) ).list()

%o A175715_list(40) # _G. C. Greubel_, Dec 04 2019

%o (GAP) a:=[1,1,2,3,8,10];; for n in [7..30] do a[n]:=a[n-1]+a[n-2]+3*a[n-4] - 4*a[n-5]+2*a[n-6]; od; a; # _G. C. Greubel_, Dec 04 2019

%K nonn,easy

%O 0,3

%A _Roger L. Bagula_, Dec 04 2010