O15-C901
> restart;with(linalg):
Warning, new definition for norm
Warning, new definition for trace
> f:=(i,j)->if j=1 then -i+1 elif i=1 then j-1 elif i=j then 1 else 0 fi;

  f := proc(i, j)
option operator, arrow;
    if j = 1 then -i + 1
    elif i = 1 then j - 1
    elif i = j then 1
    else 0
    fi
end

> a:=n->matrix(n+1,n+1,f);

                  a := n -> matrix(n + 1, n + 1, f)

> for n from 2 to 5 do [a(n),eigenvects(a(n))] od;

   [ 0    1    2]
   [            ]
  [[-1    1    0], [1, 1, {[0, -2, 1]}],
   [            ]
   [-2    0    1]

        [1/2 + 1/2 I sqrt(19), 1, {[1/2 - 1/2 I sqrt(19), 1, 2]}],

        [1/2 - 1/2 I sqrt(19), 1, {[1/2 + 1/2 I sqrt(19), 1, 2]}]]


   [ 0    1    2    3]
   [                 ]
   [-1    1    0    0]
  [[                 ], [1, 2, {[0, -3, 0, 1], [0, -2, 1, 0]}], [
   [-2    0    1    0]
   [                 ]
   [-3    0    0    1]

        1/2 + 1/2 I sqrt(55), 1,

        {[1/4 - 1/4 I sqrt(55), 1/2, 1, 3/2]}], [

        1/2 - 1/2 I sqrt(55), 1,

        {[1/4 + 1/4 I sqrt(55), 1/2, 1, 3/2]}]]


   [ 0    1    2    3    4]
   [                      ]
   [-1    1    0    0    0]
   [                      ]
  [[-2    0    1    0    0], [1/2 + 1/2 I sqrt(119), 1,
   [                      ]
   [-3    0    0    1    0]
   [                      ]
   [-4    0    0    0    1]

        {[1/8 - 1/8 I sqrt(119), 1/4, 1/2, 3/4, 1]}], [

        1/2 - 1/2 I sqrt(119), 1,

        {[1/8 + 1/8 I sqrt(119), 1/4, 1/2, 3/4, 1]}], [

        1, 3, {[0, -2, 1, 0, 0], [0, -4, 0, 0, 1], [0, -3, 0, 1, 0]}]

        ]


   [ 0    1    2    3    4    5]
   [                           ]
   [-1    1    0    0    0    0]
   [                           ]
   [-2    0    1    0    0    0]
  [[                           ], [1/2 + 1/2 I sqrt(219), 1,
   [-3    0    0    1    0    0]
   [                           ]
   [-4    0    0    0    1    0]
   [                           ]
   [-5    0    0    0    0    1]

        {[1/6 - 1/6 I sqrt(219), 1/3, 2/3, 1, 4/3, 5/3]}], [

        1/2 - 1/2 I sqrt(219), 1,

        {[1/6 + 1/6 I sqrt(219), 1/3, 2/3, 1, 4/3, 5/3]}], [1, 4, {

        [0, -3, 0, 1, 0, 0], [0, -2, 1, 0, 0, 0], [0, -5, 0, 0, 0, 1],

        [0, -4, 0, 0, 1, 0]}]]

> # Conjecture : a(n) admet 1 comme vp d'ordre n-1 et 2 nombres complexes conjugués comme vp d'ordre 1; a(n)-I_(n+1) est de rang 2 donc E_1 est de dimension n-1 et a(n) est diagonalisable
> for n from 2 to 7 do d[n]:=det(a(n)) od:d[2];[seq(d[n]-d[n-1],n=3..7)];

                                  5


                         [9, 16, 25, 36, 49]

> #Conjecture : d_n-d_(n-1)=n^2, d'où d_n=n(n+1)(2n+1)/6 (développer/dernière ligne)