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%I #14 Jul 24 2024 13:22:10
%S 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 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 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 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 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 The first few rows of the triangle are:
%e 1;
%e 4, 3;
%e 5, 4, 1;
%e 8, 7, 4, 3;
%e 9, 8, 5, 4, 1;
%e ...
%p 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 Q[i_, j_] := If[j <= i, 2 + (-1)^j, 0];
%t R[i_, j_] := If[j <= i, 1, 0];
%t T[i_, j_] := Sum[Q[i, k]*R[k, j], {k, 1, 13}];
%t Table[T[i, j], {i, 1, 13}, {j, 1, i}] // Flatten (* _Jean-François Alcover_, Jul 24 2024 *)
%Y Cf. A035608, A074377, A104570.
%Y 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 nonn,tabl
%O 1,2
%A _Gary W. Adamson_, Mar 16 2005
%E More terms from _Emeric Deutsch_, Mar 23 2005