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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.

%I A204242 #14 Dec 02 2015 11:54:40
%S A204242 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,
%T A204242 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,
%U A204242 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
%N 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.
%H A204242 Robert Israel, <a href="/A204242/b204242.txt">Table of n, a(n) for n = 1..10000</a>
%F A204242 From _Robert Israel_, Nov 30 2015: (Start)
%F A204242 a(k*(k+1)/2) = a(1 + k*(k+1)/2) = 1.
%F A204242 a(2*k^2 + 2*k + 1) = 2^(k+1) - 1.
%F A204242 a(n) = 0 otherwise. (End)
%e A204242 Northwest corner:
%e A204242 1 1 1 1
%e A204242 1 3 0 0
%e A204242 1 0 7 0
%e A204242 1 0 0 15
%p A204242 N:= 1000: # to get a(1) to a(N)
%p A204242 V:= Vector(N):
%p A204242 V[[seq(k*(k+1)/2, k= 1..floor((sqrt(8*N+1)-1)/2))]]:= 1:
%p A204242 V[[seq(1+k*(k+1)/2, k=1..floor((sqrt(8*N-7)-1)/2))]]:= 1:
%p A204242 V[[seq(1+2*k+2*k^2, k=0..floor((sqrt(2*N-1)-1)/2))]]:=
%p A204242     <seq(2^(k+1)-1,k=0..floor((sqrt(2*N-1)-1)/2))>:
%p A204242 convert(V,list); # _Robert Israel_, Nov 30 2015
%t A204242 f[i_, j_] := 0; f[1, j_] := 1; f[i_, 1] := 1; f[i_, i_] := 2^i - 1;
%t A204242 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204242 TableForm[m[8]] (* 8x8 principal submatrix *)
%t A204242 Flatten[Table[f[i, n + 1 - i],
%t A204242   {n, 1, 12}, {i, 1, n}]]     (* A204242 *)
%t A204242 Table[Det[m[n]], {n, 1, 15}]  (* A204243 *)
%t A204242 Permanent[m_] :=
%t A204242   With[{a = Array[x, Length[m]]},
%t A204242    Coefficient[Times @@ (m.a), Times @@ a]];
%t A204242 Table[Permanent[m[n]], {n, 1, 15}]   (* A203011 *)
%Y A204242 Cf. A204243, A203011.
%K A204242 nonn,tabl
%O A204242 1,5
%A A204242 _Clark Kimberling_, Jan 13 2012
%E A204242 Name edited by _Robert Israel_, Nov 30 2015