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A136493 Triangle of coefficients of characteristic polynomials of symmetrical pentadiagonal matrices of the type (1,-1,1,-1,1). 0
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Row sums are 1, 1, 0, -1, 2, 0, -2, 3, 0, -3, 4, 0 ... .

From Georg Fischer, Mar 29 2021 (Start):

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) 5x5 matrix.

(End)

REFERENCES

Anthony Ralston and Philip Rabinowitz, A First Course in Numerical Analysis, 1978, ISBN 0070511586, see p. 256.

LINKS

Table of n, a(n) for n=1..66.

EXAMPLE

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}

MATHEMATICA

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, 1, d}, {m, 1, d}];

CL[n_] := CoefficientList[CharacteristicPolynomial[MO[n], x], x];

Join[{{1}}, Table[Reverse[CL[n]], {n, 1, 10}]] // Flatten

(* For the signature of A124698 added by Georg Fischer, Mar 29 2021 : *)

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]]

CROSSREFS

Cf. A124805 ff., A124696 ff., A124999 ff.

Sequence in context: A127701 A004199 A062283 * A338849 A338838 A340108

Adjacent sequences:  A136490 A136491 A136492 * A136494 A136495 A136496

KEYWORD

tabl,sign

AUTHOR

Roger L. Bagula, Mar 21 2008

EXTENSIONS

Edited by Georg Fischer, Mar 29 2021

STATUS

approved

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Last modified October 27 22:24 EDT 2021. Contains 348305 sequences. (Running on oeis4.)