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%I #27 Sep 08 2022 08:45:12
%S 1,0,6,18,138,762,4698,27930,168090,1007514,6047130,36278682,
%T 217680282,1306065306,7836424602,47018482074,282111023514,
%U 1692665878938,10155995797914,60935973738906,365615844530586,2193695062989210
%N Expansion of (1 - 5*x - 2*x^2) / ((1 - x)*(1 + 2*x)*(1 - 6*x)).
%C Closed walks of length n at a vertex of the Johnson graph J(5,2).
%C 6^n = a(n) + 6*A091055(n) + 3*4*A091056(n).
%H Colin Barker, <a href="/A091054/b091054.txt">Table of n, a(n) for n = 0..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JohnsonGraph.html">Johnson Graph</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,8,-12).
%F a(n) = (6^n + 5*(-2)^n + 4)/10.
%F a(n) = 5*a(n-1) + 8*a(n-2) - 12*a(n-3) for n>2. - _Colin Barker_, Dec 26 2019
%F E.g.f.: (exp(6*x) + 5*exp(-2*x) + 4*exp(x))/10. - _G. C. Greubel_, Dec 27 2019
%p seq( (6^n +5*(-2)^n +4)/10, n=0..30); # _G. C. Greubel_, Dec 27 2019
%t Table[(6^n +5*(-2)^n +4)/10, {n,0,30}] (* _G. C. Greubel_, Dec 27 2019 *)
%t LinearRecurrence[{5,8,-12},{1,0,6},30] (* _Harvey P. Dale_, Oct 21 2021 *)
%o (PARI) Vec((1 - 5*x - 2*x^2) / ((1 - x)*(1 + 2*x)*(1 - 6*x)) + O(x^25)) \\ _Colin Barker_, Dec 26 2019
%o (PARI) vector(31, n, (6^(n-1) +5*(-2)^(n-1) +4)/10) \\ _G. C. Greubel_, Dec 27 2019
%o (Magma) [(6^n +5*(-2)^n +4)/10: n in [0..30]]; // _G. C. Greubel_, Dec 27 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 25); Coefficients(R!( (1 - 5*x - 2*x^2) / ((1 - x)*(1 + 2*x)*(1 - 6*x)))); // _Marius A. Burtea_, Dec 29 2019
%o (Sage) [(6^n +5*(-2)^n +4)/10 for n in (0..30)] # _G. C. Greubel_, Dec 27 2019
%o (GAP) List([0..30], n-> (6^n +5*(-2)^n +4)/10); # _G. C. Greubel_, Dec 27 2019
%Y Cf. A091055, A091056.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Dec 17 2003
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