1,8

Remember? 1/(1-x)=Sum[x^n,{n,0,Infinitity}] So to try with the Steinbach field: (I-A[i,j])^(-1)=Sun[A[i,j]^n,{n,0,Infinity}] It doesn't appear it should be finite? But I-A[i,j] is finite--> zero? {{1,0,0}, {{1,1,1}, {{0,-1,-1}, {0,1,0}, {1,1,0}, {-1,0,0}, {0,0,1}} - 1,0,0}}= { -1,0,1}} Matrices: {{0, -1}, {-1, 1}}, {{0, -1, -1}, {-1, 0, 0}, {-1, 0, 1}}, {{0, -1, -1, -1}, {-1, 0, -1, 0}, {-1, -1, 1, 0}, {-1, 0, 0, 1}}, {{0, -1, -1, -1, -1}, {-1, 0, -1, -1, 0}, {-1, -1, 0, 0, 0}, {-1, -1, 0, 1, 0}, {-1, 0, 0, 0, 1}}

P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.

I[n]-An[i,j]-> P[k,x] P[k,n]->T[n,m)

{1},

{0, -1},

{-1, -1, 1},

{-1, 2, 1, -1},

{-2, 7, -3, -2, 1},

{-1, 7, -13, 5, 2, -1},

{-1, 12, -34, 30, -6, -3, 1},

{0, 5, -30, 60, -45, 9,3, -1},

{1, 1, -41, 130, -155, 78, -10, -4, 1}

An[d_] := Table[If[n + m - 1 > d, 0, 1], {n, 1, d}, {m, 1, d}]; Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[IdentityMatrix[d] - An[d], x], x], {d, 1, 20}]]; Flatten[%]

Cf. A065941, A122767, A123218, A066170, A122610.

Sequence in context: A022874 A022873 A323739 * A058316 A082386 A336521

Adjacent sequences: A122157 A122158 A122159 * A122161 A122162 A122163

uned,tabl,sign

Gary W. Adamson and Roger L. Bagula, Oct 17 2006

approved