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.

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

Author

Roger L. Bagula, Jan 16 2008

Keywords

• nonn

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).

Programs

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 *)

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]

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