A202605 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the Fibonacci self-fusion matrix (A202453).

1, -1, 1, -3, 1, 1, -6, 9, -1, 1, -9, 26, -24, 1, 1, -12, 52, -96, 64, -1, 1, -15, 87, -243, 326, -168, 1, 1, -18, 131, -492, 1003, -1050, 441, -1, 1, -21, 184, -870, 2392, -3816, 3265, -1155, 1, 1, -24, 246, -1404, 4871, -10500, 13710

Offset

1,4

Comments

Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix. The zeros of p(n) are positive and interlace the zeros of p(n+1). (See the references and examples.)

Following is a guide to sequences (f(n)) for symmetric matrices (self-fusion matrices) and characteristic polynomials. Notation: F(k)=A000045(k) (Fibonacci numbers); floor(n*tau)=A000201(n) (lower Wythoff sequence); "periodic x,y" represents the sequence (x,y,x,y,x,y,...).

f(n)........ symmetric matrix.. char. polynomial

1............... A087062....... A202672

n............... A115262....... A202673

n^2............. A202670....... A202671

2n-1............ A202674....... A202675

3n-2............ A202676....... A202677

n(n+1)/2........ A185957....... A202678

2^n-1........... A202873....... A202767

2^(n-1)......... A115216....... A202868

floor(n*tau).... A202869....... A202870

F(n)............ A202453....... A202605

F(n+1).......... A202874....... A202875

Lucas(n)........ A202871....... A202872

F(n+2)-1........ A202876....... A202877

F(n+3)-2........ A202970....... A202971

(F(n))^2........ A203001....... A203002

(F(n+1))^2...... A203003....... A203004

C(2n,n)......... A115255....... A203005

(-1)^(n+1)...... A003983....... A076757

periodic 1,0.... A203905....... A203906

periodic 1,0,0.. A203945....... A203946

periodic 1,0,1.. A203947....... A203948

periodic 1,1,0.. A203949....... A203950

periodic 1,0,0,0 A203951....... A203952

periodic 1,2.... A203953....... A203954

periodic 1,2,3.. A203955....... A203956

...

In the cases listed above, the zeros of the characteristic polynomials are positive. If more general symmetric matrices are used, the zeros are all real but not necessarily positive - but they do have the interlace property. For a guide to such matrices and polynomials, see A202605.

Links

Table of n, a(n) for n=1..51.

S.-G. Hwang, Cauchy's interlace theorem for eigenvalues of Hermitian matrices, American Mathematical Monthly 111 (2004) 157-159.

Clark Kimberling, Fusion, Fission, and Factors, Fib. Q., 52 (2014), 195-202.

A. Mercer and P. Mercer, Cauchy's interlace theorem and lower bounds for the spectral radius, International Journal of Mathematics and Mathematical Sciences 23, no. 8 (2000) 563-566.

Example

The 1st principal submatrix (ps) of A202453 is {{1}} (using Mathematica matrix notation), with p(1) = 1-x and zero-set {1}.
...
The 2nd ps is {{1,1},{1,2}}, with p(2) = 1-3x+x^2 and zero-set {0.382..., 2.618...}.
...
The 3rd ps is {{1,1,2},{1,2,3},{2,3,6}}, with p(3) = 1-6x+9x^2-x^3 and zero-set {0.283..., 0.426..., 8.290...}.
  ...
Top of the array A202605:
  1,   -1;
  1,   -3,    1;
  1,   -6,    9,   -1;
  1,   -9,   26,  -24,    1;
  1,  -12,   52,  -96,   64,   -1;
  1,  -15,   87, -243,  326, -168,    1;

Mathematica

f[k_] := Fibonacci[k];
U[n_] := NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[f[k], {k, 1, n}]];
L[n_] := Transpose[U[n]];
F[n_] := CharacteristicPolynomial[L[n].U[n], x];
c[n_] := CoefficientList[F[n], x]
TableForm[Flatten[Table[F[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%]
TableForm[Table[c[n], {n, 1, 10}]]

Crossrefs

Cf. A000045, A202453.

Sequence in context: A158359 A046716 A371967 * A298636 A123354 A120247

Adjacent sequences: A202602 A202603 A202604 * A202606 A202607 A202608

Keyword

tabf,sign

Author

Clark Kimberling, Dec 21 2011

Status

approved