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A122160 Identity matrices minus Steinbach matrices as characteristic polynomials to give a triangular array I[n]-An[i,j]-> P[k,x] P[k,n]->T[n,m). 1
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, 1, -6, -3, 87, -220, 229, -106, 14, 4, -1, 2, -19, 45, 54, -378, 609, -455, 160, -15, -5, 1, 1, -15, 73, -123, -89, 609, -889, 615, -205, 20, 5, -1, 1, -24, 164, -460 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,8
COMMENTS
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}}
LINKS
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
FORMULA
I[n]-An[i,j]-> P[k,x] P[k,n]->T[n,m)
EXAMPLE
{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}
MATHEMATICA
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[%]
CROSSREFS
Sequence in context: A022874 A022873 A323739 * A058316 A082386 A336521
KEYWORD
uned,tabl,sign
AUTHOR
STATUS
approved

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Last modified June 21 09:49 EDT 2024. Contains 373543 sequences. (Running on oeis4.)