A134884 A three person Markov game designed to have a limiting ratio near 1/Alpha: Ma matrix=Mb = {{0, 1}, {-67, -67}}; game value =-67; Mc={{0, 1, 0}, {0, 0, 1}, {134, 0, 137}}: game value=134; Total matrix game value=-601526; Characteristic Polynomial: 601526 + 1203052 x + 1234475 x^2 + 1243453 x^3 + 624507 x^4 + 13735 x^5 + 3 x^6-x^7.

3, 1, 27269, 1954390, 385327519, 45035320119, 6682022825588, 881709781234437, 123029167626415695, 16708818725606483602, 2298930844925022134207, 314329661992199488247899, 43107655899059704928917636

Offset

1,1

Comments

Relative game value=(Game value Ma/Game value Mc)=-67/134=-1/2; Limiting ratio is: 137.00713868806855 Current value of 1/Alpha: 137.03599679 This hidden person game model is of two spin 1/2 particles interacting through a third hidden particle with a limiting constant of very near 1/Alpha.

Links

Table of n, a(n) for n=1..13.

Wikipedia, Fine Structure Constant

Index entries for linear recurrences with constant coefficients, signature (70,9112,9313,8978,8978).

Formula

M = {{0, 1, 0, 0, 0, 0, 0}, {-67, -67, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0}, {0, 0, 134, 0, 137, 0, 0}, {0, 0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 0, -67, -67}}; v[0] = {1, 0, 1, 0, 0, 0, 1}; v[n]=M.v[n-1] a(n) = Sum[v[n][[i]],{i,1,7}]

G.f.: -x*(8844*x^4 +8509*x^3 -137*x^2 -209*x +3)/((67*x^2 +67*x +1)*(134*x^3 +137*x -1)). [Colin Barker, Nov 02 2012]

Mathematica

M = {{0, 1, 0, 0, 0, 0, 0}, {-67, -67, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0}, {0, 0, 134, 0, 137, 0, 0}, {0, 0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 0, -67, -67}}; v[0] = {1, 0, 1, 0, 0, 0, 1}; v[n_] := v[n] = M.v[n - 1]; a = Table[Apply[Plus, v[n]], {n, 0, 50}]

Crossrefs

Sequence in context: A068542 A036112 A266230 * A229850 A269162 A033909

Adjacent sequences: A134881 A134882 A134883 * A134885 A134886 A134887

Keyword

nonn,uned,easy

Author

Roger L. Bagula, Jan 29 2008

Status

approved