1,1

Matrices: {{4}}, {{4, -1}, {-1, 4}}, {{4, -1, 0}, {-1, 4, -1}, {0, -1, 4}}, {{4, -1, 0, 0}. {-1, 4, -1, 0}, {0, -1, 4, -1}, {0, 0, -1, 4}}, {{4, -1, 0, 0, 0}, {-1, 4, -1, 0, 0}, {0, -1, 4, -1, 0}, {0, 0, -1, 4, -1}, {0, 0, 0, -1, 4}}, {{4, -1, 0, 0, 0, 0}, {-1, 4, -1, 0, 0, 0}, {0, -1, 4, -1, 0, 0}, {0, 0, -1, 4, -1, 0}, {0, 0, 0, -1, 4, -1}, {0, 0, 0, 0, -1, 4}}

Eric Weisstein's World of Mathematics, "Tridiagonal Matrix." http://mathworld.wolfram.com/TridiagonalMatrix.html

a(n,m)=If[ n == m, 4, If[n == m - 1 || n == m + 1, -1, 0]]] p(n,x)=CharacteristicPolynomial(a(n,m)) p(n,x)->t(n,m)

Polynomials:

4

4 - x,

15 - 8 x + x^2,

56 - 46 x + 12 x^2 - x^3,

209 - 232 x + 93 x^2 - 16 x^3 + x^4,

780 - 1091 x + 592 x^2 - 156 x^3 + 20 x^4 - x^5,

2911 - 4912 x + 3366 x^2 - 1200 x^3 + 235 x^4 - 24 x^5 + x^6

Triangular sequence:

{4},

{4, -1},

{15, -8, 1},

{56, -46,12, -1},

{209, -232, 93, -16, 1},

{780, -1091, 592, -156, 20, -1},

{2911, -4912, 3366, -1200, 235, -24, 1},

{10864, -21468, 17784, -8010, 2120, -330, 28, -1}

Clear[M, T, d, a, x]; T[n_, m_] = If[ n == m, 4, If[n == m - 1 || n == m + 1, -1, 0]]; M[d_] := Table[T[n, m], {n, 1, d}, {m, 1, d}]; Table[M[d], {d, 1, 10}]; Table[Det[M[d] - x*IdentityMatrix[d]], {d, 1, 10}]; a = Join[{{3}}, Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], {d, 1, 10}]]; Flatten[a]

Cf. A123343.

Sequence in context: A140313 A102323 A124028 this_sequence A079507 A098364 A116866

Adjacent sequences: A123963 A123964 A123965 this_sequence A123967 A123968 A123969

uned,probation,tabl,sign

Gary Adamson and Roger Bagula (rlbagulatftn(AT)yahoo.com), Oct 28 2006

Looking at the triangle suggests that the very first term should be 1, not 4. - njas, Nov 01, 2006