A202868 -id:A202868 - cp's OEIS frontend

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

Clark Kimberling, Dec 21 2011

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.

			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;

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.

Cf. A000045, A202453.

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}]]

A115216 "Correlation triangle" for 2^n.

Original entry on oeis.org

1, 2, 2, 4, 5, 4, 8, 10, 10, 8, 16, 20, 21, 20, 16, 32, 40, 42, 42, 40, 32, 64, 80, 84, 85, 84, 80, 64, 128, 160, 168, 170, 170, 168, 160, 128, 256, 320, 336, 340, 341, 340, 336, 320, 256, 512, 640, 672, 680, 682, 682, 680, 672, 640, 512, 1024, 1280, 1344, 1360, 1364

Offset: 0

Paul Barry, Jan 16 2006

Row sums are A102301. T(2n,n) gives A002450(n+1). Diagonal sums are A115217.
Construction: Take antidiagonal triangle of MM^T where M is the sequence array for the sequence 2^n.
When formated as a square array, this is the self-fusion matrix (as in Example and Mathematica sections) of the sequence (2^n); for interlacing zeros of associated characteristic polynomials, see A202868.  [Clark Kimberling, Dec 26 2011]

			Triangle begins
  1,
  2, 2,
  4, 5, 4,
  8, 10, 10, 8,
  16, 20, 21, 20, 16,
  32, 40, 42, 42, 40, 32,
  ...
Northwest corner of square matrix:
  1....2....4....8....16
  2....5....10...20...40
  4....10...21...42...85
  8....20...41...85...170
  16...40...84...170..341
  ..

Cf. A003983, A202678.

(* A115216 as a square matrix *)
s[k_] := 2^(k - 1);
U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 12}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]
Table[f[n], {n, 1, 12}]
Table[Sqrt[f[n]], {n, 1, 12}]  (* -1+2^n *)
Table[m[n, n], {n, 1, 12}]  (* A002450 *)
(* Clark Kimberling, Dec 26 2011 *)

T(n, k) = Sum_{j=0..n} [j<=k]*2^(k-j)[j<=n-k]*2^(n-k-j).

G.f.: 1/((1-2*x)*(1-2*x*y)*(1-x^2*y)). - Christian G. Bower, Jan 17 2006

cp's OEIS Frontend

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

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