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 A134186 A 3 person 9 X 9 Markov approach to a zero sum game where: Sum[game_value(MAi),{i,1,3}]=0 and two of the games are minimal Pisot vector Markovs and the third is a negative Fibonacci: Characteristic Polynomial: -1 + 3 x^2 + 3 x^3 - 4 x^4 - 5 x^5 + x^6 + 4 x^7 - x^9; MA1={{0,1,0},{,0,0,1},{1,1,0}};Det=1 ;gv=-1/4; MA2={{0,1,1},{1,0,0},{0,1,0}};Det=1;gv=-1/4 MA2={{0,0,1},{0,1,0},{1,0,-1}};Det=-1;gv=1/2. 0
 4, 3, 5, 4, 7, 7, 12, 11, 23, 16, 45, 21, 90, 19, 187, -14, 405, -149, 912, -587, 2127, -1882, 5111, -5533, 12578, -15549, 31521, -42574, 80051, -114735, 205238, -306127, 529707, -811536, 1373431, -2142327, 3572244, -5639743, 9311113, -14819542, 24304609, -38893711, 63503588, -101992905 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS score sum Ratio approaches=-(1+Sqrt[5])/2 Score is low positive to the 16th element then starts to alternate. The negative Fibonacci game is being balanced by the two Minimal Pisot games. Roots are: Table[x /. NSolve[CharacteristicPolynomial[M, x] == 0, x][[n]], {n, 1, 9}]; {-1.61803, -0.662359 - 0.56228 I, -0.662359 + 0.56228 I, -0.662359 - 0.56228 I, -0.662359 + 0.56228 I], 0.618034, 1., 1.32472, 1.32472} LINKS Index entries for linear recurrences with constant coefficients, signature (-1,2,2,0,-1). FORMULA M = {{0, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, 0}, {1, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 1, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 1, 0, -1}}; v[0] = {0, 0, 1, 1, 0, 0, 1, 0, 1}; v[n_] := v[n] = M.v[n - 1]; a(n) = Sum[v[n][[i]],{,i,1,9}]. G.f.: -x*(5*x^4+5*x^3-7*x-4)/((x^2-x-1)*(x^3+x^2-1)). [Colin Barker, Nov 01 2012] MATHEMATICA M = {{0, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, 0}, {1, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 1, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 1, 0, -1}}; v[0] = {0, 0, 1, 1, 0, 0, 1, 0, 1}; v[n_] := v[n] = M.v[n - 1]; a = Table[Apply[Plus, v[n]], {n, 0, 50}] CROSSREFS Sequence in context: A177033 A104569 A093619 * A024688 A024477 A270649 Adjacent sequences:  A134183 A134184 A134185 * A134187 A134188 A134189 KEYWORD uned,sign,easy AUTHOR Roger L. Bagula, Jan 13 2008 STATUS approved

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Last modified May 14 16:48 EDT 2021. Contains 343898 sequences. (Running on oeis4.)