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A129686 Triangle read by rows: row n is 0^(n-3), 1, 0, 1.

%I A129686 #12 Feb 20 2022 01:25:41
%S A129686 1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0,0,1,0,1,0,0,0,0,1,0,1,0,0,0,0,0,1,
%T A129686 0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,1,0,0,
%U A129686 0,0,0,0,0,0,0,1,0,1
%N A129686 Triangle read by rows: row n is 0^(n-3), 1, 0, 1.
%C A129686 Alternate term operator, sums.
%C A129686 Let A129686 = matrix M, with V any sequence as a vector. Then M*V is the alternate term sum operator. Given V = [1,2,3,...], M*V = [1, 2, 4, 6, 8, 10, 12, 14, ...]. The analogous operation using A097807, (the pairwise operator), gives [1, 3, 5, 7, 9, 11, 13, 15, ...]. Binomial transform of A129686 = A124725. A129686 * A007318 = A129687. Row sums of A129686 = (1, 1, 2, 2, 2, ...).
%F A129686 As an infinite lower triangular matrix, (1,1,1,...) in the main diagonal, (0,0,0,...) in the subdiagonal and (1,1,1,...) in the subsubdiagonal; with the rest zeros. (1, 0, 1, 0, 0, 0, ...) in every column.
%e A129686 First few rows of the triangle:
%e A129686   1;
%e A129686   0, 1;
%e A129686   1, 0, 1;
%e A129686   0, 1, 0, 1
%e A129686   0, 0, 1, 0, 1;
%e A129686   0, 0, 0, 1, 0, 1;
%e A129686   ...
%t A129686 T[n_, k_] := If[k == n || k == n-2, 1, 0];
%t A129686 Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jul 10 2019 *)
%o A129686 (PARI) tabl(nn) = {t129686 = matrix(nn, nn, n, k, (k<=n)*((k==n) || (k==(n-2)))); for (n = 1, nn, for (k = 1, n, print1(t129686[n, k], ", ");););} \\ _Michel Marcus_, Feb 12 2014
%Y A129686 Cf. A124725, A129687.
%K A129686 nonn,tabl
%O A129686 1,1
%A A129686 _Gary W. Adamson_, Apr 28 2007
%E A129686 More terms from _Michel Marcus_, Feb 12 2014