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Search: id:A124029
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| A124029 |
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Two methods for producing a fours center tridiagonal based triangulat sequence: 1) Matrix characteristic polynomial ( first element 4 instead of 1): M(n,m,d)=If[ n == m, k, If[n == m - 1 || n == m + 1, -1, 0]];k=4; 2) recursive polynomial( as here):a(k)=-1;b(k)=4; p(k, x) = ((x - b(k - 1))*p(k - 1, x) - a(k - 2) *p(k - 2, x))/a(k - 1);. |
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| 1, 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, 40545, -91824, 89238, -48624, 16255, -3416, 441, -32, 1, 151316, -386373, 430992, -275724, 111524, -29589, 5152, -568, 36
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OFFSET
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1,2
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REFERENCES
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Joanne Dombrowski, Tridiagonal matrix representations of cyclic selfadjoint operators, Pacific J. Math. 114, no. 2 (1984), 325-334
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FORMULA
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a(k)=-1;b(k)=4; p(k, x) = ((x - b(k - 1))*p(k - 1, x) - a(k - 2) *p(k - 2, x))/a(k - 1); M(n,m,d)=If[ n == m, k, If[n == m - 1 || n == m + 1, -1, 0]];k=4;
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EXAMPLE
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Triangular sequence:
{1},--> {4} in the matrix version
{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}
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MATHEMATICA
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(* Matrix version*) k = 4; T[n_, m_, d_] := If[ n == m, k, If[n == m - 1 || n == m + 1, -1, 0]] M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}] Table[M[d], {d, 1, 10}] Table[Det[M[d]], {d, 1, 10}] Table[Det[M[d] - x*IdentityMatrix[d]], {d, 1, 10}] a = Join[{M[1]}, Table[CoefficientList[Det[M[ d] - x*IdentityMatrix[d]], x], {d, 1, 10}]] Flatten[a] MatrixForm[a] (* Recursive Polynomial form*) b[k_] = 4; a[k_] = -1; p[0, x] = 1; p[1, x] = (x - b[1])/a[1]; p[k_, x_] := p[k, x] = ((x - b[k - 1])*p[k - 1, x] - a[k - 2] *p[k - 2, x])/a[k - 1; w = Table[CoefficientList[p[n, x], x], {n, 0, 10}]; Flatten[w]
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CROSSREFS
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Cf. A123966.
Sequence in context: A107873 A080419 A095307 this_sequence A056920 A123382 A059222
Adjacent sequences: A124026 A124027 A124028 this_sequence A124030 A124031 A124032
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KEYWORD
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uned,probation,sign
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AUTHOR
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Gary Adamson and Roger Bagula (rlbagulatftn(AT)yahoo.com), Nov 01 2006
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