A135839 Triangle read by rows: starting with A128174, replace left border with (1, 1, 1, ...).

1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1

Offset

1,1

Comments

Row sums = (1, 2, 2, 3, 3, 4, 4, 5, 5, ...).

Is this (apart from the offset) a duplicate of A071022? - R. J. Mathar, Mar 03 2026

Links

Zhuorui He, Rows 1..150 of the triangle, flattened (first 50 rows from G. C. Greubel)

Formula

Given A128174, replace left border with (1, 1, 1, ...). Triangle read by rows, odd rows = n terms of (1, 0, 1, ...); even rows = n terms of (1, 1, 0, 1, 0, 1, ...). By columns, leftmost column = (1, 1, 1, ...); all others = (1, 0, 1, 0, 1, ...).

T(n,1) = T(n,n) = 1, T(n,k) = (1 - (-1)^(n-k-1))/2. - G. C. Greubel,Dec 05 2016

Example

First few rows of the triangle are:
  1;
  1, 1;
  1, 0, 1;
  1, 1, 0, 1;
  1, 0, 1, 0, 1;
  1, 1, 0, 1, 0, 1;
  1, 0, 1, 0, 1, 0, 1;
  ...

Mathematica

T[1, 1] := 1; T[n_, 1] := 1; T[n_, n_] := 1; T[n_, k_] := (1 - (-1)^(n - k + 1))/2; Table[T[n, k], {n, 1, 10}, {k, 1, n}]//Flatten (* G. C. Greubel, Dec 05 2016 *)
Flatten[Table[Join[{1}, PadLeft[{}, n, {0, 1}]], {n, 0, 20}]] (* Harvey P. Dale, Feb 26 2024 *)

Crossrefs

Sequence in context: A103994 A051731 A304569 * A071022 A155076 A328308

Adjacent sequences: A135836 A135837 A135838 * A135840 A135841 A135842

Keyword

nonn,tabl,easy

Author

Gary W. Adamson, Dec 01 2007

Extensions

Definition corrected by Zhuorui He, Jul 21 2025

Status

approved