A135647 G.f. = 1/(x^36*p(1/x)) where p(x)=(- 25 - 49 x^9 + x^10)*(- 1 - 2 x^9 + x^10)^3*(- 1 - x^9 + x^10)^6.

1, 61, 3070, 150836, 7392650, 362245994, 17750074048, 869753690956, 42617931038803, 2088278621406591, 102325652450274784, 5013956970066973919, 245683891533290673468, 12038510685131268747080, 589887023571432406862284

Offset

1,2

Comments

Weighted solution of a zero sum game.

Let Ma={{0, 1, 0, 0, 0, 0, 0, 0, 0, 0},

{0, 0, 1, 0, 0, 0, 0, 0, 0, 0},

{0, 0, 0, 1, 0, 0, 0, 0, 0, 0},

{0, 0, 0, 0, 1, 0, 0, 0, 0, 0},

{0, 0, 0, 0, 0, 1, 0, 0, 0, 0},

{0, 0, 0, 0, 0, 0, 1, 0, 0, 0},

{0, 0, 0, 0, 0, 0, 0, 1, 0, 0},

{0, 0, 0, 0, 0, 0, 0, 0, 1, 0},

{0, 0, 0, 0, 0, 0, 0, 0, 0, 1},

{25, 0, 0, 0, 0, 0, 0, 0, 0, 49}}; a={1,2};

ML={{0, 1, 0, 0, 0, 0, 0, 0, 0, 0},

{0, 0, 1, 0, 0, 0, 0, 0, 0, 0},

{0, 0, 0, 1, 0, 0, 0, 0, 0, 0},

{0, 0, 0, 0, 1, 0, 0, 0, 0, 0},

{0, 0, 0, 0, 0, 1, 0, 0, 0, 0},

{0, 0, 0, 0, 0, 0, 1, 0, 0, 0},

{0, 0, 0, 0, 0, 0, 0, 1, 0, 0},

{0, 0, 0, 0, 0, 0, 0, 0, 1, 0},

{0, 0, 0, 0, 0, 0, 0, 0, 0, 1},

{25, 0, 0, 0, 0, 0, 0, 0, 0, 49}}.

Such that:

6*Game_value[M1]+3*Game_value[M2]+Game_Value[ML]=0

My first solution was "unweighted".

Links

Table of n, a(n) for n=1..15.

Formula

p(x)=(-25 - 49 x^9 + x^10)(-1 - 2 x^9 + x^10)^3(-1 - x^9 + x^10)^6; f(x)=1/(x^36*p(1/x)) a(n) =expansion(f(x))

Mathematica

f[x_] = Product[CharacteristicPolynomial[{{0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {1, 0, 0, 0, 0, 0, 0, 0, 0, a}}, x]^(6/a), {a, 1, 2}]*CharacteristicPolynomial[{{0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {25, 0, 0, 0, 0, 0, 0, 0, 0, 49}}, x]; g[x_] = Expand[x^100*f[1/x]]; a = Table[ SeriesCoefficient[Series[1/g[x], {x, 0, 30}], n], {n, 0, 30}]

Crossrefs

Sequence in context: A000508 A191092 A234028 * A375893 A269025 A207231

Adjacent sequences: A135644 A135645 A135646 * A135648 A135649 A135650

Keyword

nonn,uned,obsc

Author

Roger L. Bagula, Jan 31 2008

Extensions

The connection with the zero-sum game is not clear to me. Also, how does Ma depend on a? It appears that Ma = ML, so perhaps there are errors in these matrices? - N. J. A. Sloane, May 16 2008

Status

approved