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A320975 Greatest eigenvalues of the form x + y*sqrt(z) of the Fibonacci matrix of the form n X n = [[F(0),F(1),...,F(n-1)],[F(n),F(n+1),...,F(2n-1)],...,[(F(n(n-1)),F(n(n-1)+1),...,F(n^2-1)]] represented by the triples of integers (x, y, z). 0
0, 0, 0, 1, 1, 2, 12, 2, 39, 335, 2, 28129, 24552, 2, 150705906, 4778306, 2, 5708053098753, 2456043408, 52, 2230824417240678, 3322443961836, 3, 1226514875504825822765861, 11804340768687390, 4, 8708903811455950754101585556070 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

We consider an important property of the matrix n X n whose elements are the Fibonacci numbers, because this matrix has only two nonzero eigenvalues of the form lambda1 = x + y*sqrt(z) > 0, lambda2 = x - y*sqrt(z) < 0 for n > 1. The other eigenvalues are all 0 (with multiplicity n-2).

Property:

lambda1 + lambda2 = floor(lambda1) = trace (n X n), where trace (n X n) is defined by the sum of the elements on the main diagonal. This property is checked for a large values of n.

For n > 1, trace(n X n) = 2, 24, 670,... = A174997(n).

We observe that y*sqrt(z) - x tends into 1/sqrt(5) when n is infinite.

We give a formulation of the n X n matrix where F_k is the k-th Fibonacci number, as follows:

n = 1 =>   [F_0] = [0]

n = 2 =>   [F_0, F_1]  = [0,  1]

           [F_2, F_3]    [1,  2]

n = 3 =>   [F_0, F_1, F_2]    [0,  1,  1]

           [F_3, F_4, F_5]  = [2,  3,  5]

           [F_6, F_7, F_8]    [8, 13, 21]

..........................................

The following table gives the first eigenvalues for n = 2, 3, 4, 5 (the

case n = 1 is not listed in the table).

+---+--------------------------+---------------------------+-----------+

| n |   positive eigenvalues   |    negative eigenvalues   |eigenvalues|

|   |                          |                           |equal to 0 |

+---+--------------------------+---------------------------+-----------+

| 2 |     1 + sqrt(2)          |     1 - sqrt(2)           |           |

| 3 |    12 + 2*sqrt(39)       |    12 - 2*sqrt(39)        |0          |

| 4 |   335 + 2*sqrt(28129)    |   335 - 2*sqrt(28129)     |0, 0       |

| 5 | 24552 + 2*sqrt(150705906)| 24552 - 2*sqrt(150705906) |0, 0, 0    |

.......................................................................

REFERENCES

G. H. Golub and C. F. van Loan, Matrix Computations, Johns Hopkins, 1989, p. 336.

Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001.

LINKS

Table of n, a(n) for n=1..27.

S. L. Basin and Verner E. Hoggatt Jr., A Primer on the Fibonacci Sequence--Part II, Fib. Quart. 1, 61-68, 1963.

J. L. Brenner, June Meeting of the Pacific Northwest Section. 1. Lucas' Matrix, Amer. Math. Monthly 58, 220-221, 1951.

Graham Fisher, The Singularity of Fibonacci Matrices, Mathematical Gazette, Vol. 81 (1997), 295-298.

Eric Weisstein's World of Mathematics, Fibonacci Q-Matrix

MAPLE

with(linalg):with(combinat, fibonacci):

for n from 1 to 9 do:

m:=n*n:T:=array(0..m-1):A:=matrix(n, n, T):

  for k from 0 to n^2-1 do :

   T[k]:=fibonacci(k):

  od:

  print(eigenvalues(A)) :

od:

MATHEMATICA

sqMax[n_] := If[n == 1, 1, Times@@Power@@@({#[[1]], #[[2]] - Mod[#[[2]], 2]} & /@ FactorInteger [n])]; a[n_] := If[n==0, {0, 0, 0}, Module[{v = Select[ Eigenvalues[Table[Fibonacci[i*(n+1)+j], {i, 0, n}, {j, 0, n}]], #!=0 &]}, s = Simplify[ Total@v ]/2; p = s^2 - Simplify[Times@@v]; sqr=sqMax[p]; {s, Sqrt[ sqr], p/sqr}]]; Array[a, 10, 0] //Flatten (* Amiram Eldar, Dec 16 2018 *)

CROSSREFS

Cf. A000045, A174997.

Sequence in context: A277265 A263631 A010239 * A222804 A128268 A185633

Adjacent sequences:  A320972 A320973 A320974 * A320976 A320977 A320978

KEYWORD

nonn

AUTHOR

Michel Lagneau, Dec 16 2018

STATUS

approved

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Last modified May 18 10:19 EDT 2021. Contains 343995 sequences. (Running on oeis4.)