cp's OEIS Frontend

A134326 The sum of the elements in the first, middle and last row of the n-th power of the 9-by-9 matrix defined in the formula.

%I A134326 #12 Mar 06 2022 15:44:00
%S A134326 3,11,34,112,359,1167,3764,12191,39391,127434,411973,1332290,4307638,
%T A134326 13928919,45036841,145621921,470842799,1522389829,4922341763,
%U A134326 15915370482,51458800352,166380151440,537950254595,1739329494378
%N A134326 The sum of the elements in the first, middle and last row of the n-th power of the 9-by-9 matrix defined in the formula.
%C A134326 The characteristic polynomial is -5 x - 8 x^2 + 42 x^3 + 37 x^4 - 19 x^5 - 23 x^6 + 5 x^7 + 4 x^8 - x^9.
%C A134326 The largest root of the polynomial is 3.23322.
%C A134326 The value of the associated matrix game is zero.
%H A134326 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (4,5,-23,-19,37,42,-8,-5).
%F A134326 M = {{0, 1, 0, 1, 0, 0, 1, 0, 0}, {0, 0, 1, 0, 1, 0, 0, 1, 0}, {1, 0, 0, 0, 0, 1, 0, 0, 1}, {1, 0, 0, 0, 1, 0, 1, 0, 0}, {0, 1, 0, 1, 0, 0, 0, 1, 0}, {0, 0, 1, 0, 1, 0, 0, 0, 1}, {1, 0, 0, 1, 0, 0, 1, 1, 0}, {0, 1, 0, 0, 1, 0, 0, 1, 0}, {0, 0, 1, 0, 0, 1, 0, 0, 2}}; v[0] = {1, 0, 0, 0, 1, 0, 0, 0, 1}; v[n_] := v[n] = M.v[n - 1]; a(n) = Sum[v[n][[i]],{i,1,9}]
%F A134326 a(n)= 4*a(n-1) +5*a(n-2) -23*a(n-3) -19*a(n-4) +37*a(n-5) +42*a(n-6) -8*a(n-7) -5*a(n-8). G.f.: -x*(-3+x+25*x^2+10*x^3-51*x^4-51*x^5+10*x^6+11*x^7) / (1-4*x-5*x^2+23*x^3+ 19*x^4-37*x^5-42*x^6+8*x^7+5*x^8). [From _R. J. Mathar_, Aug 12 2009]
%t A134326 M = {{0, 1, 0, 1, 0, 0, 1, 0, 0}, {0, 0, 1, 0, 1, 0, 0, 1, 0}, {1, 0, 0, 0, 0, 1, 0, 0, 1}, {1, 0, 0, 0, 1, 0, 1, 0, 0}, {0, 1, 0, 1, 0, 0, 0, 1, 0}, {0, 0, 1, 0, 1, 0, 0, 0, 1}, {1, 0, 0, 1, 0, 0, 1, 1, 0}, {0, 1, 0, 0, 1, 0, 0, 1, 0}, {0, 0, 1, 0, 0, 1, 0, 0, 2}};
%t A134326 v[0] = {1, 0, 0, 0, 1, 0, 0, 0, 1}; v[n_] := v[n] = M.v[n - 1]; a = Table[Apply[Plus, v[n]], {n, 0, 50}]
%t A134326 LinearRecurrence[{4,5,-23,-19,37,42,-8,-5},{3,11,34,112,359,1167,3764,12191},30] (* _Harvey P. Dale_, Mar 06 2022 *)
%K A134326 nonn
%O A134326 0,1
%A A134326 _Roger L. Bagula_, Jan 16 2008
%E A134326 Offset set to zero by the Associate Editors of the OEIS, Sep 11 2009
%E A134326 Linear recurrence index corrected by _Harvey P. Dale_, Mar 06 2022