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%I #28 Oct 16 2013 06:11:02
%S 5,4,12,25,64,159,411,1068,2808,7423,19717,52529,140251,375015,
%T 1003770,2688570,7204696,19313075,51782613,138861732,372414289,
%U 998851473,2679146955,7186319506,19276417059,51707411684,138702360471,372064319188
%N Expansion of (5-16*x+6*x^2+10*x^3-2*x^4)/(1-4*x+2*x^2+5*x^3-2*x^4-x^5)
%C Same as A062883 preceded by 5.
%C Let U be the unit-primitive matrix (see [Jeffery])
%C U=U_(11,2)=
%C (0 0 1 0 0)
%C (0 1 0 1 0)
%C (1 0 1 0 1)
%C (0 1 0 1 1)
%C (0 0 1 1 1).
%C Then a(n)=Trace(U^n).
%C Evidently one of a class of accelerator sequences for Catalan's constant based on traces of successive powers of a unit-primitive matrix U_(N,r) (0<r<floor(N/2)) and for which the closed-form expression for a(n) is derived from the eigenvalues of U_(N,r).
%C Formulae given below are special cases of general one's defined and discussed in Witula-Slota's paper. For example a(n) = A(n;1), where A(n;d) := Sum_{k=1..5} (1 + 2d*cos(2Pi*k/11))^n, n=0,1,..., d in C. - _Roman Witula_, Jul 26 2012
%D R. Witula and D. Slota, Quasi-Fibonacci Numbers of Order 11, 10 (2007), J. Integer Seq., Article 07.8.5.
%H L. E. Jeffery, <a href="/wiki/User:L._Edson_Jeffery/Unit-Primitive_Matrices">Unit-primitive matrices</a>
%F G.f.: (5-16*x+6*x^2+10*x^3-2*x^4)/(1-4*x+2*x^2+5*x^3-2*x^4-x^5).
%F a(n)=4*a(n-1)-2*a(n-2)-5*a(n-3)+2*a(n-4)+a(n-5), {a(m)}=5,4,12,25,64, m=0..4.
%F a(n)=Sum_{k=1..5} ((x_k)^2-1)^n; x_k=2*(-1)^(k-1)*cos(k*Pi/11).
%t u = {{0, 0, 1, 0, 0}, {0, 1, 0, 1, 0}, {1, 0, 1, 0, 1}, {0, 1, 0, 1, 1}, {0, 0, 1, 1, 1}}; a[n_] := Tr[ MatrixPower[u, n] ]; Table[a[n], {n, 0, 27}] (* _Jean-François Alcover_, Oct 14 2013 *)
%o (PARI) Vec((5-16*x+6*x^2+10*x^3-2*x^4)/(1-4*x+2*x^2+5*x^3-2*x^4-x^5)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 25 2012
%Y Cf. A189234, A189236, A189237.
%K nonn,easy
%O 0,1
%A _L. Edson Jeffery_, Apr 18 2011
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