A204242 Infinite symmetric matrix given by f(i,1)=1, f(1,j)=1, f(i,i)=2^i-1 and f(i,j)=0 otherwise, read by antidiagonals.

1, 1, 1, 1, 3, 1, 1, 0, 0, 1, 1, 0, 7, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 15, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 31, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 63, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 127, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0

Offset

1,5

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

From Robert Israel, Nov 30 2015: (Start)

a(k*(k+1)/2) = a(1 + k*(k+1)/2) = 1.

a(2*k^2 + 2*k + 1) = 2^(k+1) - 1.

a(n) = 0 otherwise. (End)

Example

Northwest corner:
1 1 1 1
1 3 0 0
1 0 7 0
1 0 0 15

Maple

N:= 1000: # to get a(1) to a(N)
V:= Vector(N):
V[[seq(k*(k+1)/2, k= 1..floor((sqrt(8*N+1)-1)/2))]]:= 1:
V[[seq(1+k*(k+1)/2, k=1..floor((sqrt(8*N-7)-1)/2))]]:= 1:
V[[seq(1+2*k+2*k^2, k=0..floor((sqrt(2*N-1)-1)/2))]]:=
    <seq(2^(k+1)-1, k=0..floor((sqrt(2*N-1)-1)/2))>:
convert(V, list); # Robert Israel, Nov 30 2015

Mathematica

f[i_, j_] := 0; f[1, j_] := 1; f[i_, 1] := 1; f[i_, i_] := 2^i - 1;
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8x8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 12}, {i, 1, n}]]     (* A204242 *)
Table[Det[m[n]], {n, 1, 15}]  (* A204243 *)
Permanent[m_] :=
  With[{a = Array[x, Length[m]]},
   Coefficient[Times @@ (m.a), Times @@ a]];
Table[Permanent[m[n]], {n, 1, 15}]   (* A203011 *)

Crossrefs

Cf. A204243, A203011.

Sequence in context: A115718 A361510 A204181 * A211313 A368847 A321609

Adjacent sequences: A204239 A204240 A204241 * A204243 A204244 A204245

Keyword

nonn,tabl

Author

Clark Kimberling, Jan 13 2012

Extensions

Name edited by Robert Israel, Nov 30 2015

Status

approved