A133566 - OEIS

%I #24 Jul 24 2024 02:09:58

%S 1,0,1,0,1,1,0,0,0,1,0,0,0,1,1,0,0,0,0,0,1,0,0,0,0,0,1,1,0,0,0,0,0,0,

%T 0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,0,0,

%U 0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1

%N Triangle read by rows: (1,1,1,...) on the main diagonal and (0,1,0,1,...) on the subdiagonal.

%C Usually regarded as a square matrix T when combined with other matrices and column vectors.

%C Then T * V, where V = any sequence regarded as a column vector with offset 1 is a new sequence S [called an interpolation transform] given by S(2n) = V(2n), S(2n-1) = V(2n) + V(2n-1). Example: If T * [1,2,3,...], S = [1, 2, 5, 4, 9, 6, 13, 8, 17, ...) = A114752. A133080 is identical to A133566 except that the subdiagonal = (1,0,1,0,...). A133080 * [1,2,3,...] = A114753: (1, 3, 3, 7, 5, 11, 7, 15, 9, 19, ...).

%C Triangle T(n,k), 0 <= k <= n, read by rows given by [0,1,-1,0,0,0,0,0,0,...] DELTA [1,0,-2,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Dec 15 2007

%F Odd rows: (n-2) zeros followed by 1, 1. Even rows: (n-1) zeros followed by 1.

%F Sum_{k=0..n} T(n,k) = A040001(n). - _Philippe Deléham_, Dec 15 2007

%F G.f.: (-1-x*y-x^2*y)*x*y/((-1+x*y)*(1+x*y)). - _R. J. Mathar_, Aug 11 2015

%e First few rows of the triangle:

%e   1;

%e   0, 1;

%e   0, 1, 1;

%e   0, 0, 0, 1;

%e   0, 0, 0, 1, 1;

%e   0, 0, 0, 0, 0, 1;

%e   ...

%p A133566 := proc(n,k)

%p     if n = k then

%p         1;

%p     elif  k=n-1 and type(n,odd) then

%p         1;

%p     else

%p         0 ;

%p     end if;

%p end proc: # _R. J. Mathar_, Jun 20 2015

%t T[n_, k_] := Which[n == k, 1, k == n - 1 && OddQ[n], 1, True, 0];

%t Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Aug 24 2023 *)

%Y Cf. A133080, A114752, A114753, A084938, A040001.

%K nonn,tabl,easy

%O 1,1

%A _Gary W. Adamson_, Sep 16 2007

%E Entry revised by _N. J. A. Sloane_, Jun 20 2015