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%I #20 Mar 26 2024 06:04:16
%S 1,4,18,74,309,1280,5313,22035,91410,379171,1572857,6524375,27063881,
%T 112264055,465684247,1931711700,8012962189,33238687760,137877896315,
%U 571933356551,2372445281505,9841175633650,40822327332150,169335704473650,702423959724591
%N Let M denote the 6 X 6 matrix with rows /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) = z(n).
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,6,-4,-5,1,1).
%F G.f.: -(x+1) / (x^6+x^5-5*x^4-4*x^3+6*x^2+3*x-1). - _Colin Barker_, Jun 14 2013
%p a:= n->(Matrix(6, (i, j)->`if`(i+j>7, 0, 1))^n.<<[1$6][]>>)[3, 1]:
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Jun 14 2013
%t m = Table[ If[i + j <= 7, 1, 0], {i, 1, 6}, {j, 1, 6}]; mp[n_] := MatrixPower[m, n].m[[1]]; a[n_] := mp[n][[3]]; Table[a[n], {n, 0, 24}] (* _Jean-François Alcover_, Jun 18 2013 *)
%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 _Benoit Cloitre_, Apr 02 2002
%E Edited by _Henry Bottomley_, May 06 2002