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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
 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 (list; table; graph; refs; listen; history; text; internal format)
 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 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: A137251 A158359 A046716 * A298636 A123354 A120247 Adjacent sequences:  A202602 A202603 A202604 * A202606 A202607 A202608 KEYWORD tabl,sign AUTHOR Clark Kimberling, Dec 21 2011 STATUS approved

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Last modified October 20 23:28 EDT 2020. Contains 337910 sequences. (Running on oeis4.)