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A070778
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Let M denote the 6 X 6 matrix = row by row /1,1,1,1,1,1/1,1,1,1,1,0/1,1,1,1,0,0/1,1,1,0,0,0/1,1,0,0,0,0/1,0,0,0,0,0/ and A(n) the vector (x(n),y(n),z(n),t(n),u(n),v(n)) = M^n*A where A is the vector (1,1,1,1,1,1); then a(n) = u(n).
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%I #18 Sep 08 2022 08:45:06
%S 1,2,11,41,176,721,3003,12439,51623,214103,888173,3684174,15282475,
%T 63393324,262962987,1090800411,4524765831,18769248040,77856998326,
%U 322959774150,1339674254489,5557122741105,23051583675890,95620617831960,396645310086831,1645330322871807
%N Let M denote the 6 X 6 matrix = row by row /1,1,1,1,1,1/1,1,1,1,1,0/1,1,1,1,0,0/1,1,1,0,0,0/1,1,0,0,0,0/1,0,0,0,0,0/ and A(n) the vector (x(n),y(n),z(n),t(n),u(n),v(n)) = M^n*A where A is the vector (1,1,1,1,1,1); then a(n) = u(n).
%H Vincenzo Librandi, <a href="/A070778/b070778.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 (3,6,-4,-5,1,1).
%F a(n) = 2*A006359(n-1) - A006359(n-3) for n > 2.
%F G.f.: (x^2 + x - 1) / (x^6 + x^5 - 5*x^4 - 4*x^3 + 6*x^2 + 3*x - 1). - _Colin Barker_, Jun 14 2013
%F a(n) = 3*a(n-1) + 6*a(n-2) - 4*a(n-3) - 5*a(n-4) + a(n-5) + a(n-6). - _Wesley Ivan Hurt_, Oct 09 2017
%p a:= n-> (Matrix(6, (i, j)->`if`(i+j>7, 0, 1))^n.<<[1$6][]>>)[5, 1]:
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Jun 14 2013
%t CoefficientList[Series[(x^2 + x - 1)/(x^6 + x^5 - 5*x^4 - 4*x^3 + 6*x^2 + 3*x - 1), {x, 0, 30}], x] (* _Wesley Ivan Hurt_, Oct 09 2017 *)
%t LinearRecurrence[{3, 6, -4, -5, 1, 1}, {1, 2, 11, 41, 176, 721}, 30] (* _Vincenzo Librandi_, Oct 10 2017 *)
%o (Magma) I:=[1,2,11,41,176,721]; [n le 6 select I[n] else 3*Self(n-1)+6*Self(n-2)-4*Self(n-3)-5*Self(n-4)+Self(n-5)+Self(n-6): n in [1..30]]; // _Vincenzo Librandi_, Oct 10 2017
%Y Cf. A006359, A069007, A069008, A069009, A070778, A006359 (offset), for x(n), y(n), z(n), t(n), u(n), v(n).
%K easy,nonn
%O 0,2
%A _Henry Bottomley_, May 06 2002
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