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A202605 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the Fibonacci self-fusion matrix (A202453). 78

%I

%S 1,-1,1,-3,1,1,-6,9,-1,1,-9,26,-24,1,1,-12,52,-96,64,-1,1,-15,87,-243,

%T 326,-168,1,1,-18,131,-492,1003,-1050,441,-1,1,-21,184,-870,2392,

%U -3816,3265,-1155,1,1,-24,246,-1404,4871,-10500,13710

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

%C 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.)

%C 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,...).

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

%C 1............... A087062....... A202672

%C n............... A115262....... A202673

%C n^2............. A202670....... A202671

%C 2n-1............ A202674....... A202675

%C 3n-2............ A202676....... A202677

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

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

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

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

%C F(n)............ A202453....... A202605

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

%C Lucas(n)........ A202871....... A202872

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

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

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

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

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

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

%C periodic 1,0.... A203905....... A203906

%C periodic 1,0,0.. A203945....... A203946

%C periodic 1,0,1.. A203947....... A203948

%C periodic 1,1,0.. A203949....... A203950

%C periodic 1,0,0,0 A203951....... A203952

%C periodic 1,2.... A203953....... A203954

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

%C ...

%C 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.

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

%H S.-G. Hwang, <a href="http://matrix.skku.ac.kr/Series-E/Monthly-E.pdf">Cauchy's interlace theorem for eigenvalues of Hermitian matrices</a>, American Mathematical Monthly 111 (2004) 157-159.

%H A. Mercer and P. Mercer, <a href="http://dx.doi.org/10.1155/S016117120000257X">Cauchy's interlace theorem and lower bounds for the spectral radius</a>, International Journal of Mathematics and Mathematical Sciences 23, no. 8 (2000) 563-566.

%e The 1st principal submatrix (ps) of A202453 is {{1}} (using Mathematica matrix notation), with p(1) = 1-x and zero-set {1}.

%e ...

%e The 2nd ps is {{1,1},{1,2}}, with p(2) = 1-3x+x^2 and zero-set {0.382..., 2.618...}.

%e ...

%e 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...}.

%e ...

%e Top of the array A202605:

%e 1, -1;

%e 1, -3, 1;

%e 1, -6, 9, -1;

%e 1, -9, 26, -24, 1;

%e 1, -12, 52, -96, 64, -1;

%e 1, -15, 87, -243, 326, -168, 1;

%t f[k_] := Fibonacci[k];

%t U[n_] := NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[f[k], {k, 1, n}]];

%t L[n_] := Transpose[U[n]];

%t F[n_] := CharacteristicPolynomial[L[n].U[n], x];

%t c[n_] := CoefficientList[F[n], x]

%t TableForm[Flatten[Table[F[n], {n, 1, 10}]]]

%t Table[c[n], {n, 1, 12}]

%t Flatten[%]

%t TableForm[Table[c[n], {n, 1, 10}]]

%Y Cf. A000045, A202453.

%K tabl,sign

%O 1,4

%A _Clark Kimberling_, Dec 21 2011

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Last modified November 14 15:24 EST 2019. Contains 329126 sequences. (Running on oeis4.)