%I A122573
%S A122573 1,1,1,1,3,3,11,11,41,41,153,153,571,571,2131,2131,7953,7953,29681,
%T A122573 29681,110771,110771,413403,413403,1542841,1542841,5757961,5757961,
%U A122573 21489003,21489003,80198051,80198051,299303201,299303201,1117014753
%N A122573 4 X 4 vector matrix Markov for characteristic polynomial:(1 - 4 x^2 + 
               x^4) Which Mathematica gives as a factor of the cubic polynomial: 
               Factor[(1 - 14 x^4 + x^8)]=(1 - 4 x^2 + x^4)(1 + 4 x^2 + x^4) Also:a(n)=4*a(n-2)-a(n-4).
%C A122573 A001835[n]=v[2*n][[1]]=4*a[n-1]-a[n-2]: The coefficient expansion gives 
               an alternating even term zeros sequence: p[x_] := x^4 - 4x^2 + 1 
               q[x_] := ExpandAll[x^4*p[1/x]] Table[ SeriesCoefficient[Series[x/
               q[x], {x, 0, 30}], n], {n, 0, 30}] {0, 1, 0, 4, 0, 15, 0, 56, 0, 
               209, 0, 780, 0, 2911, 0, 10864, 0, 40545, 0, 151316, 0, 564719, 0, 
               2107560, 0, 7865521, 0, 29354524, 0, 109552575, 0}
%F A122573 M = {{0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {-1, 0, 4, 0}}; v[1] = 
               {1, 1, 1, 1}; v[n_] := v[n] = M.v[n - 1] a(n) = v[n][[1]]
%t A122573 M = {{0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {-1, 0, 4, 0}}; v[1] = 
               {1, 1, 1, 1}; v[n_] := v[n] = M.v[n - 1] a1 = Table[v[n][[1]], {n, 
               1, 50}] (* alternative calculation method*) a[0] = 1; a[1] = 1; a[2] 
               = 1; a[3] = 1; a[n_] := a[n] = 4*a[n - 2] - a[n - 4] Table[a[n], 
               {n, 0, 50}]
%Y A122573 Cf. A001835.
%Y A122573 Sequence in context: A146828 A146583 A146458 this_sequence A136123 A045495 
               A045494
%Y A122573 Adjacent sequences: A122570 A122571 A122572 this_sequence A122574 A122575 
               A122576
%K A122573 nonn,uned
%O A122573 1,5
%A A122573 Roger Bagula (rlbagulatftn(AT)yahoo.com), Sep 17 2006