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A136493
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Triangle of coefficients of characteristic polynomials of symmetrical pentadiagonal matrices of the type (1,-1,1,-1,1).
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1
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1, -1, 1, 1, -2, 0, -1, 3, 0, 0, 1, -4, 1, 2, 0, -1, 5, -3, -5, 1, 1, 1, -6, 6, 8, -5, -2, 1, -1, 7, -10, -10, 14, 4, -4, 0, 1, -8, 15, 10, -29, -4, 12, 0, 0, -1, 9, -21, -7, 50, -4, -30, 4, 4, 0, 1, -10, 28, 0, -76, 28, 61, -20, -15, 2, 1
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OFFSET
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0,5
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COMMENTS
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The pentadiagonal matrices have 1 in the main diagonal, -1 in the first lower and upper diagonal, 1 in the second lower and upper diagonal, and 0 otherwise.
The linear recurrences that yield A124805, A124806, A124807 and similar can be derived from the rows of this triangle (the first element of a row must be removed and multiplied onto the remaining elements).
This observation extends to other sequences. For example the linear recurrence signature (5,-6,2,4,0) of A124698 "Number of base 5 circular n-digit numbers with adjacent digits differing by 1 or less" can be derived from the coefficients of the characteristic polynomial of a tridiagonal (type -1,1,-1) 5 X 5 matrix.
(End)
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REFERENCES
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Anthony Ralston and Philip Rabinowitz, A First Course in Numerical Analysis, 1978, ISBN 0070511586, see p. 256.
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LINKS
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FORMULA
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Sum_{k=1..n} T(n, k) = (-1)^(n mod 3) * A087509(n+1) + [n=1].
T(n, 1) = (-1)^(n-1).
T(n, 3) = (-1)^(n-3)*A161680(n-3). (End)
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EXAMPLE
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Triangle begins:
1;
-1, 1;
1, -2, 0;
-1, 3, 0, 0;
1, -4, 1, 2, 0;
-1, 5, -3, -5, 1, 1;
1, -6, 6, 8, -5, -2, 1;
-1, 7, -10, -10, 14, 4, -4, 0;
1, -8, 15, 10, -29, -4, 12, 0, 0;
-1, 9, -21, -7, 50, -4, -30, 4, 4, 0;
1, -10, 28, 0, -76, 28, 61, -20, -15, 2, 1;
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MATHEMATICA
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T[n_, m_]:= Piecewise[{{-1, 1+m==n || m==1+n}, {1, 2+m==n || m==n || m==2+n}}];
MO[d_]:= Table[T[n, m], {n, d}, {m, d}];
CL[n_]:= CoefficientList[CharacteristicPolynomial[MO[n], x], x];
Join[{{1}}, Table[Reverse[CL[n]], {n, 10}]]//Flatten
Reverse[CoefficientList[CharacteristicPolynomial[{{1, -1, 0, 0, 0}, {-1, 1, -1, 0, 0}, {0, -1, 1, -1, 0}, {0, 0, -1, 1, -1}, {0, 0, 0, -1, 1}}, x], x]]
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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