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A134704
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"Hidden" person three person game Markov: MA=MB={{0.1},{1,1}}: Fibonacci: game value =1 MC={{-2, -2, 0}, {1, 0, 1}, {0, 1, 1}}: game value=-2 A plays with C; B plays with C; but A has no direct contact with B . A, B are the "observed" games. Characteristic polynomial: 4 + 9 x - 3 x^2 - 12 x^3 + x^4 + 4 x^5 + x^6 - x^7.
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0
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3, 3, 10, 9, 17, 40, 25, 103, 114, 141, 469, 264, 989, 1507, 1114, 5281, 3369, 8808, 19633, 9039, 56162, 48133, 71245, 245320, 89301, 555627, 700426, 512505, 2896001, 1203880, 5013385, 9831415, 3325650, 31944381, 19420165, 39925128, 130084109
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OFFSET
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1,1
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COMMENTS
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C is the "dark" or "hidden" player. it is simpler than a three three by three games, but the middle matrix was hard to find at gv=-2. Roots: aaa = Table[x /. NSolve[Det[M - x*IdentityMatrix[7]] == 0, x][[n]], {n, 1, 7}]; {-1.24279 - 1.07145 I, -1.24279 + 1.07145I, -0.618034, -0.618034, 1.48558, 1.61803, 1.61803} Ratio: a1 = Table[N[a[[n]]/a[[n - 1]]], {n, 7, 50}]; The ratio Alternates. Game value of total matrix is -1: Det[M]/(Sum[Sum[If[i == j, M[[i, j]], 0], {i, 1, 7}], {j, 1, 7}] - Sum[Sum[If[i == j, 0, M[[i, j]]], {i, 1, 7}], {j, 1, 7}])
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LINKS
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FORMULA
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M = {{0, 1, 0, 0, 0, 0, 0}, {1, 1, 0, 0, 0, 0, 0}, {0, 0, -2, -2, 0, 0, 0}, {0, 0, 1, 0, 1, 0, 0}, {0, 0, 0, 1, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 1, 1}}; v[1] = {0, 1, 0, 1, 0, 0, 1}; v[n]=M.v[n-1]; a(n) =Sum[v[n][[i],{i,1,7}]
G.f.: -x*(10*x^4+12*x^3-x^2-3*x-3)/((x^2+x-1)*(4*x^3+x^2-x-1)). [Colin Barker, Nov 01 2012]
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EXAMPLE
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mc = {{-2, -2, 0}, {1, 0, 1}, {0, 1, 1}};
Game Value mc=Det[mc]/(Sum[Sum[If[i == j, mc[[i, j]], 0], {i, 1, 3}], {j, 1, 3}] - Sum[Sum[If[i == j, 0, mc[[i, j]]], {i, 1, 3}], {j, 1, 3}])=-2
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MATHEMATICA
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M = {{0, 1, 0, 0, 0, 0, 0}, {1, 1, 0, 0, 0, 0, 0}, {0, 0, -2, -2, 0, 0, 0}, {0, 0, 1, 0, 1, 0, 0}, {0, 0, 0, 1, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 1, 1}}; v[1] = {0, 1, 0, 1, 0, 0, 1}; v[n_] := v[n] = M.v[n - 1]; a = Table[Apply[Plus, v[n]], {n, 1, 50}] Det[M - x*IdentityMatrix[7]]; Factor[%]
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CROSSREFS
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KEYWORD
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nonn,uned,easy
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AUTHOR
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STATUS
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approved
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