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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.
1
3, 11, 34, 112, 359, 1167, 3764, 12191, 39391, 127434, 411973, 1332290, 4307638, 13928919, 45036841, 145621921, 470842799, 1522389829, 4922341763, 15915370482, 51458800352, 166380151440, 537950254595, 1739329494378
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.
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
Sequence in context: A037496 A355364 A180762 * A295546 A361435 A094308
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