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A123965 Triangle read by rows: T(0,0)=1; T(n,k)=coefficient of x^k in the polynomial (-1)^n*p[n,x], where p[n,x] is the monic characteristic polynomial of the n X n tridiagonal matrix with 3's on the main diagonal and -1's on the super- and subdiagonal (n>=1; 0<=k<=n). 5
1, 3, -1, 8, -6, 1, 21, -25, 9, -1, 55, -90, 51, -12, 1, 144, -300, 234, -86, 15, -1, 377, -954, 951, -480, 130, -18, 1, 987, -2939, 3573, -2305, 855, -183, 21, -1, 2584, -8850, 12707, -10008, 4740, -1386, 245, -24, 1, 6765, -26195, 43398, -40426, 23373, -8715, 2100, -316, 27, -1, 17711, -76500, 143682 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

T(n,0)=fibonacci(2n+2)=A001906(n+1).

Reversed polynomials = bisection of A152063: (1; 1,3; 1,6,8; 1,9,25,21;...) having the following property: even indexed Fibonacci numbers = Product_{k=1..(n-2/2)} (1 + 4*cos^2 k*Pi/n); n relating to regular polygons with an even number of edges. Example: The roots to x^3 - 9x^2 + 25x - 21 relate to the octagon and are such that the product with k=1,2,3 = (4.414213...,)*(3)*(1.585786...,) = 21. - Gary W. Adamson, Aug 15 2010

REFERENCES

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

LINKS

Table of n, a(n) for n=0..57.

FORMULA

a(n,m)=If[ n == m, 3, If[n == m - 1 || n == m + 1, -1, 0]]] p(n,x)=CharacteristicPolynomial(a(n,m)) p(n,x)->t(n,m)

EXAMPLE

Polynomials:

1

3 - x,

8 - 6 x + x^2,

21 - 25x + 9 x^2 - x^3,

55 - 90x + 51 x^2 - 12 x^3 + x^4,

144 - 300x + 234x^2 - 86x^3 + 15x^4 - x^5,

377 - 954 x + 951 x^2 - 480 x^3 + 130x^4 - 18 x^5 + x^6,

987 - 2939 x + 3573x^2 - 2305 x^3 + 855 x^4 - 183 x^5 + 21 x^6 - x^7

Triangular sequence:

{1},

{3, -1},

{8, -6, 1},

{21, -25, 9, -1},

{55, -90, 51, -12, 1},

{144, -300,234, -86, 15, -1},

{377, -954, 951, -480, 130, -18, 1},

{987, -2939, 3573, -2305, 855, -183, 21, -1},

{2584, -8850, 12707, -10008, 4740, -1386, 245, -24, 1},

{6765, -26195, 43398, -40426, 23373, -8715, 2100, -316, 27, -1}

MAPLE

with(linalg): a:=proc(i, j) if j=i then 3 elif abs(i-j)=1 then -1 else 0 fi end: for n from 1 to 10 do p[n]:=(-1)^n*charpoly(matrix(n, n, a), x) od: 1; for n from 1 to 10 do seq(coeff(p[n], x, j), j=0..n) od; # yields sequence in triangular form

MATHEMATICA

Clear[M, T, d, a, x]; T[n_, m_] = If[ n == m, 3, 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]

CROSSREFS

Cf. A123343.

Cf. A001906.

Cf. A152063. - Gary W. Adamson, Aug 15 2010

Sequence in context: A030523 A207815 A125662 * A124025 A257488 A286416

Adjacent sequences:  A123962 A123963 A123964 * A123966 A123967 A123968

KEYWORD

sign,tabl

AUTHOR

Gary W. Adamson and Roger L. Bagula, Oct 28 2006

EXTENSIONS

Edited by N. J. A. Sloane, Nov 24 2006

STATUS

approved

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Last modified July 17 02:10 EDT 2019. Contains 325092 sequences. (Running on oeis4.)