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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).

%I A202605 #35 Feb 17 2025 03:18:58
%S 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,
%T A202605 326,-168,1,1,-18,131,-492,1003,-1050,441,-1,1,-21,184,-870,2392,
%U A202605 -3816,3265,-1155,1,1,-24,246,-1404,4871,-10500,13710
%N A202605 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the Fibonacci self-fusion matrix (A202453).
%C A202605 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 A202605 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 A202605 f(n)........ symmetric matrix.. char. polynomial
%C A202605 1............... A087062....... A202672
%C A202605 n............... A115262....... A202673
%C A202605 n^2............. A202670....... A202671
%C A202605 2n-1............ A202674....... A202675
%C A202605 3n-2............ A202676....... A202677
%C A202605 n(n+1)/2........ A185957....... A202678
%C A202605 2^n-1........... A202873....... A202767
%C A202605 2^(n-1)......... A115216....... A202868
%C A202605 floor(n*tau).... A202869....... A202870
%C A202605 F(n)............ A202453....... A202605
%C A202605 F(n+1).......... A202874....... A202875
%C A202605 Lucas(n)........ A202871....... A202872
%C A202605 F(n+2)-1........ A202876....... A202877
%C A202605 F(n+3)-2........ A202970....... A202971
%C A202605 (F(n))^2........ A203001....... A203002
%C A202605 (F(n+1))^2...... A203003....... A203004
%C A202605 C(2n,n)......... A115255....... A203005
%C A202605 (-1)^(n+1)...... A003983....... A076757
%C A202605 periodic 1,0.... A203905....... A203906
%C A202605 periodic 1,0,0.. A203945....... A203946
%C A202605 periodic 1,0,1.. A203947....... A203948
%C A202605 periodic 1,1,0.. A203949....... A203950
%C A202605 periodic 1,0,0,0 A203951....... A203952
%C A202605 periodic 1,2.... A203953....... A203954
%C A202605 periodic 1,2,3.. A203955....... A203956
%C A202605 ...
%C A202605 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.
%H A202605 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 A202605 Clark Kimberling, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/52-3/Kimberling11132013.pdf">Fusion, Fission, and Factors</a>, Fib. Q., 52 (2014), 195-202.
%H A202605 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 A202605 The 1st principal submatrix (ps) of A202453 is {{1}} (using Mathematica matrix notation), with p(1) = 1-x and zero-set {1}.
%e A202605 ...
%e A202605 The 2nd ps is {{1,1},{1,2}}, with p(2) = 1-3x+x^2 and zero-set {0.382..., 2.618...}.
%e A202605 ...
%e A202605 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 A202605   ...
%e A202605 Top of the array A202605:
%e A202605   1,   -1;
%e A202605   1,   -3,    1;
%e A202605   1,   -6,    9,   -1;
%e A202605   1,   -9,   26,  -24,    1;
%e A202605   1,  -12,   52,  -96,   64,   -1;
%e A202605   1,  -15,   87, -243,  326, -168,    1;
%t A202605 f[k_] := Fibonacci[k];
%t A202605 U[n_] := NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[f[k], {k, 1, n}]];
%t A202605 L[n_] := Transpose[U[n]];
%t A202605 F[n_] := CharacteristicPolynomial[L[n].U[n], x];
%t A202605 c[n_] := CoefficientList[F[n], x]
%t A202605 TableForm[Flatten[Table[F[n], {n, 1, 10}]]]
%t A202605 Table[c[n], {n, 1, 12}]
%t A202605 Flatten[%]
%t A202605 TableForm[Table[c[n], {n, 1, 10}]]
%Y A202605 Cf. A000045, A202453.
%K A202605 tabf,sign
%O A202605 1,4
%A A202605 _Clark Kimberling_, Dec 21 2011