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

Author

Clark Kimberling, Jan 13 2012

Keywords

• nonn
• tabl

Examples

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

Links

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

Crossrefs

Cf. A204243, A203011.

Programs

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))]]:=
    :
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 *)

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)

Extensions

Name edited by Robert Israel, Nov 30 2015