A106465 Triangle read by rows, T(n, k) = 1 if n mod 2 = 1, otherwise (k + 1) mod 2.

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

Offset

0,1

Comments

Rows alternate between all 1's and alternating 1's and 0's. A 'mixed' sequence array: rows alternate between the rows of the sequence array for the all 1's sequence and the sequence array for the sequence 1,0,1,0,...

Column 2*k has g.f. x^(2*k)/(1-x); column 2*k+1 has g.f. x^(2*k+1)/(1-x^2).

Row sums are A029578(n+2). Antidiagonal sums are A106466.

This triangle is the Kronecker product of an infinite lower triangular matrix filled with 1's with a 2 X 2 lower triangular matrix of 1's. - Christopher Cormier, Sep 24 2017

From Peter Bala, Aug 21 2021: (Start)

Using the notation of Davenport et al.:

This is the double Riordan array ( 1/(1 - x); x/(1 + x), x*(1 + x) ).

The inverse array equals ( (1 - x)*(1 - x^2); x*(1 - x), x*(1 + x) ).

They are examples of double Riordan arrays of the form (g(x); x*f_1(x), x*f_2(x)), where f_1(x)*f_2(x) = 1. Arrays of this type form a group under matrix multiplication. For the group law see the Bala link. (End)

Links

Table of n, a(n) for n=0..104.

Peter Bala, Matrices with repeated columns - the generalised Appell groups

D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012).

Formula

If gcd(n - k + 1, k + 1) mod 2 = 0 then T(n, k) = 0, otherwise T(n, k) = 1.

T(n, k) = A003989(n + 1, k + 1) mod 2.

T(n, k) = binomial(n mod 2, k mod 2). - Peter Luschny, Dec 12 2022

Example

The triangle begins:
  n\k| 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 ...
  ---+------------------------------------------------
   0 | 1
   1 | 1  1
   2 | 1  0  1
   3 | 1  1  1  1
   4 | 1  0  1  0  1
   5 | 1  1  1  1  1  1
   6 | 1  0  1  0  1  0  1
   7 | 1  1  1  1  1  1  1  1
   8 | 1  0  1  0  1  0  1  0  1
   9 | 1  1  1  1  1  1  1  1  1  1
  10 | 1  0  1  0  1  0  1  0  1  0  1
  11 | 1  1  1  1  1  1  1  1  1  1  1  1
  12 | 1  0  1  0  1  0  1  0  1  0  1  0  1
  13 | 1  1  1  1  1  1  1  1  1  1  1  1  1  1
  14 | 1  0  1  0  1  0  1  0  1  0  1  0  1  0  1
... Reformatted by Wolfdieter Lang, May 12 2018
Inverse array begins
  n\k|  0   1   2   3   4   5   6   7
  ---+-------------------------------
   0 |  1
   1 | -1   1
   2 | -1   0   1
   3 |  1  -1  -1   1
   4 |  0   0  -1   0   1
   5 |  0   0   1  -1  -1   1
   6 |  0   0   0   0  -1   0   1
   7 |  0   0   0   0   1  -1  -1  1
  ... - Peter Bala, Aug 21 2021

Maple

T := (n, k) -> if igcd(n - k + 1, k + 1) mod 2 = 0 then 0 else 1 fi:
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;
# Alternative:
T := (n, k) -> if n mod 2 = 1 then 1 else (k + 1) mod 2 fi:
for n from 0 to 9 do seq(T(n, k), k = 0..n) od; # Peter Luschny, Dec 12 2022

Mathematica

Table[Binomial[Mod[n, 2], Mod[k, 2]], {n, 0, 16}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 12 2022 *)

Prog

(Python)
def A106465row(n: int) -> list[int]:
  if n % 2 == 1:
      return [1] * (n + 1)
  return [1, 0] * (n // 2) + [1]
for n in range(9): print(A106465row(n)) # Peter Luschny, Dec 12 2022

Crossrefs

Cf. A003989, A029578, A106466.

Sequence in context: A054431 A164381 A106470 * A071027 A337264 A099990

Adjacent sequences: A106462 A106463 A106464 * A106466 A106467 A106468

Keyword

easy,nonn,tabl

Author

Paul Barry, May 03 2005

Extensions

Edited and new name by Peter Luschny, Dec 12 2022

Status

approved