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%I #17 Sep 08 2022 08:45:28
%S 0,1,5,40,315,2490,19681,155563,1229604,9719061,76821600,607214857,
%T 4799560053,37936780428,299860673343,2370164848026,18734305316497,
%U 148080078051971,1170457572108040,9251554605638681,73126326541645648,578006601205833441
%N The (1,5)-entry in the 5 X 5 matrix M^n, where M={{5, 3, 2, 1, 1}, {3, 2, 1, 1, 0}, {2, 1, 1, 0, 0}, {1, 1, 0, 0, 0}, {1, 0, 0, 0, 0}}.
%D See A123942 for references.
%H Colin Barker, <a href="/A123943/b123943.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (8,0,-6,0,1).
%F a(n) = 8*a(n-1) - 6*a(n-3) + a(n-5) for n>=5 (follows from the minimal polynomial of the matrix M).
%F G.f.: x*(1 - 3*x + x^3) / (1 - 8*x + 6*x^3 - x^5). - _Colin Barker_, Mar 03 2017
%p with(linalg): M[1]:=matrix(5,5,[5,3,2,1,1,3,2,1,1,0,2,1,1,0,0,1,1,0,0,0,1,0,0,0,0]): for n from 2 to 30 do M[n]:=multiply(M[1],M[n-1]) od: 0,seq(M[n][1,5],n=1..30);
%p a[0]:=0: a[1]:=1: a[2]:=5: a[3]:=40: a[4]:=315: for n from 5 to 30 do a[n]:=8*a[n-1]-6*a[n-3]+a[n-5] od: seq(a[n],n=0..30);
%p # third Maple program:
%p a:= n-> (<<0|1|0|0|0>, <0|0|1|0|0>, <0|0|0|1|0>, <0|0|0|0|1>,
%p <1|0|-6|0|8>>^n. <<0, 1, 5, 40, 315>>)[1, 1]:
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Aug 05 2019
%t M = {{5,3,2,1,1}, {3,2,1,1,0}, {2,1,1,0,0}, {1,1,0,0,0}, {1,0,0,0,0}}; v[1] = {0,0,0,0,1}; v[n_]:= v[n] = M.v[n-1]; Table[v[n][[1]], {n, 30}]
%t LinearRecurrence[{8,0,-6,0,1}, {0,1,5,40,315}, 30] (* _G. C. Greubel_, Aug 05 2019 *)
%o (PARI) concat(0, Vec(x*(1-3*x+x^3)/(1-8*x+6*x^3-x^5) + O(x^30))) \\ _Colin Barker_, Mar 03 2017
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!( x*(1-3*x+x^3)/(1-8*x+6*x^3-x^5) )); // _G. C. Greubel_, Aug 05 2019
%o (Sage) (x*(1-3*x+x^3)/(1-8*x+6*x^3-x^5)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Aug 05 2019
%o (GAP) a:=[0,1,5,40,315];; for n in [6..30] do a[n]:=8*a[n-1]-6*a[n-3] +a[n-5]; od; a; # _G. C. Greubel_, Aug 05 2019
%Y Cf. A122099, A122100.
%K nonn,easy
%O 0,3
%A _Roger L. Bagula_ and _Gary W. Adamson_, Oct 25 2006
%E Edited by _N. J. A. Sloane_, Dec 04 2006
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