|
%I #13 Aug 11 2019 01:13:41
%S 1,2,-1,0,-4,1,2,-9,8,-1,-2,-3,19,-12,1,-4,-6,47,-55,18,-1,2,15,0,-88,
%T 93,-24,1,2,23,-7,-190,324,-182,32,-1,0,-12,-63,62,332,-554,274,-40,1,
%U 2,-9,-108,133,678,-1642,1346,-450,50,-1,-2,-11,55,276,-463,-1129,2832,-2128,630,-60,1,-4,-30,71,543,-1044,-2204,7761
%N Triangular array from Steinbach matrices plus their squares as characteristic polynomials: M[i,j]=A[i,j]+A[i,j]^2: A[i,j]^2=Min[i,j]; CharacteristicPolynomial[M[i,j]];.
%C Based on the idea that the Steinbach matrices form a "golden Field". Matrices are: {{2, 2}, {2, 2}}, {{2, 2, 2}, {2, 3, 2}, {2, 2, 3}}, {{2, 2, 2, 2}, {2, 3, 3, 2}, {2, 3, 3, 3}, {2, 2, 3, 4}}, {{2, 2, 2, 2, 2}, {2, 3, 3, 3, 2}, {2, 3, 4, 3, 3}, {2, 3, 3, 4, 4}, {2, 2, 3, 4, 5}}, {{2, 2, 2, 2, 2, 2}, {2,3, 3, 3, 3, 2}, {2, 3, 4, 4, 3, 3}, {2, 3, 4, 4, 4, 4}, {2, 3, 3, 4, 5, 5}, {2, 2, 3, 4, 5, 6}}
%H P. Steinbach, <a href="https://www.jstor.org/stable/2691048">Golden fields: a case for the heptagon</a>, Math. Mag. 70 (1997), no. 1, 22-31.
%F d-th level M(i,j)->An[d]; T(n,m)=CoefficientList[CharacteristicPolynomial[An[d], x], x]
%e {1},
%e {2, -1},
%e {0, -4, 1},
%e {2, -9, 8, -1},
%e {-2, -3, 19, -12, 1},
%e {-4, -6,47, -55, 18, -1}
%e {2, 15, 0, -88, 93, -24, 1},
%e {2, 23, -7, -190, 324, -182, 32, -1},
%e {0, -12, -63, 62, 332, -554, 274, -40, 1}
%t An[d_] := Table[Min[n, m] + If[n + m - 1 > d, 0, 1], {n, 1, d}, {m, 1, d}]; Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[An[d], x], x], {d,1, 20}]]; Flatten[%]
%Y Cf. A038223, A038223, A076756, A054142.
%K tabl,uned,sign
%O 1,2
%A _Gary W. Adamson_ and _Roger L. Bagula_, Oct 16 2006
|
