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A104569 Triangle read by rows: T(i,j) is the (i,j)-entry (1 <= j <= i) of the product Q*R of the infinite lower triangular matrices Q = [1; 1,3; 1,3,1; 1 3,1,3; ...] and R = [1; 1,1; 1,1,1; 1,1,1,1; ...].

%I A104569 #14 Jul 24 2024 13:22:10
%S A104569 1,4,3,5,4,1,8,7,4,3,9,8,5,4,1,12,11,8,7,4,3,13,12,9,8,5,4,1,16,15,12,
%T A104569 11,8,7,4,3,17,16,13,12,9,8,5,4,1,20,19,16,15,12,11,8,7,4,3,21,20,17,
%U A104569 16,13,12,9,8,5,4,1,24,23,20,19,16,15,12,11,8,7,4,3,25,24,21,20,17,16,13,12,9,8,5,4,1
%N A104569 Triangle read by rows: T(i,j) is the (i,j)-entry (1 <= j <= i) of the product Q*R of the infinite lower triangular matrices Q = [1; 1,3; 1,3,1; 1 3,1,3; ...] and R = [1; 1,1; 1,1,1; 1,1,1,1; ...].
%F A104569 For 1<=j<=i: T(i, j)=2(i-j+1) if i and j are of opposite parity; T(i, j)=2(i-j)+1 if both i and j are odd; T(i, j)=2(i-j)+3 if both i and j are even. - _Emeric Deutsch_, Mar 23 2005
%e A104569 The first few rows of the triangle are:
%e A104569   1;
%e A104569   4, 3;
%e A104569   5, 4, 1;
%e A104569   8, 7, 4, 3;
%e A104569   9, 8, 5, 4, 1;
%e A104569   ...
%p A104569 T:=proc(i,j) if j>i then 0 elif i+j mod 2 = 1 then 2*(i-j)+2 elif i mod 2 = 1 and j mod 2 = 1 then 2*(i-j)+1 elif i mod 2 = 0 and j mod 2 = 0 then 2*(i-j)+3 else fi end: for i from 1 to 13 do seq(T(i,j),j=1..i) od; # yields sequence in triangular form # _Emeric Deutsch_, Mar 23 2005
%t A104569 Q[i_, j_] := If[j <= i, 2 + (-1)^j, 0];
%t A104569 R[i_, j_] := If[j <= i, 1, 0];
%t A104569 T[i_, j_] := Sum[Q[i, k]*R[k, j], {k, 1, 13}];
%t A104569 Table[T[i, j], {i, 1, 13}, {j, 1, i}] // Flatten (* _Jean-François Alcover_, Jul 24 2024 *)
%Y A104569 Cf. A035608, A074377, A104570.
%Y A104569 Row sums yield A074377. Columns 1, 3, 5, ... (starting at the diagonal entry) yield A042948. Columns 2, 4, 6, ... (starting at the diagonal entry) yield A014601. The product R*Q yields A104570.
%K A104569 nonn,tabl
%O A104569 1,2
%A A104569 _Gary W. Adamson_, Mar 16 2005
%E A104569 More terms from _Emeric Deutsch_, Mar 23 2005