OFFSET
0,1
COMMENTS
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.
The largest root of the polynomial is 3.23322.
The value of the associated matrix game is zero.
LINKS
Index entries for linear recurrences with constant coefficients, signature (4,5,-23,-19,37,42,-8,-5).
FORMULA
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}]
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]
MATHEMATICA
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 = Table[Apply[Plus, v[n]], {n, 0, 50}]
LinearRecurrence[{4, 5, -23, -19, 37, 42, -8, -5}, {3, 11, 34, 112, 359, 1167, 3764, 12191}, 30] (* Harvey P. Dale, Mar 06 2022 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Roger L. Bagula, Jan 16 2008
EXTENSIONS
Offset set to zero by the Associate Editors of the OEIS, Sep 11 2009
Linear recurrence index corrected by Harvey P. Dale, Mar 06 2022
STATUS
approved