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A135647
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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.
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0
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1, 61, 3070, 150836, 7392650, 362245994, 17750074048, 869753690956, 42617931038803, 2088278621406591, 102325652450274784, 5013956970066973919, 245683891533290673468, 12038510685131268747080, 589887023571432406862284
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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".
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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))
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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}]
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CROSSREFS
| Sequence in context: A078962 A000508 A191092 * A207011 A206975 A206969
Adjacent sequences: A135644 A135645 A135646 * A135648 A135649 A135650
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KEYWORD
| nonn,uned,obsc
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 31 2008
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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 (njas(AT)research.att.com), May 16 2008
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