A106465 - OEIS

%I #55 Mar 21 2023 15:32:21

%S 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,

%T 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,

%U 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

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

%C 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,...

%C 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).

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

%C 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

%C From _Peter Bala_, Aug 21 2021: (Start)

%C Using the notation of Davenport et al.:

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

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

%C 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)

%H Peter Bala, <a href="/A177994/a177994.pdf">Matrices with repeated columns - the generalised Appell groups</a>

%H D. E. Davenport, L. W. Shapiro and L. C. Woodson, <a href="https://doi.org/10.37236/2034">The Double Riordan Group</a>, The Electronic Journal of Combinatorics, 18(2) (2012).

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

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

%F T(n, k) = binomial(n mod 2, k mod 2). - _Peter Luschny_, Dec 12 2022

%e The triangle begins:

%e   n\k| 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 ...

%e   ---+------------------------------------------------

%e    0 | 1

%e    1 | 1  1

%e    2 | 1  0  1

%e    3 | 1  1  1  1

%e    4 | 1  0  1  0  1

%e    5 | 1  1  1  1  1  1

%e    6 | 1  0  1  0  1  0  1

%e    7 | 1  1  1  1  1  1  1  1

%e    8 | 1  0  1  0  1  0  1  0  1

%e    9 | 1  1  1  1  1  1  1  1  1  1

%e   10 | 1  0  1  0  1  0  1  0  1  0  1

%e   11 | 1  1  1  1  1  1  1  1  1  1  1  1

%e   12 | 1  0  1  0  1  0  1  0  1  0  1  0  1

%e   13 | 1  1  1  1  1  1  1  1  1  1  1  1  1  1

%e   14 | 1  0  1  0  1  0  1  0  1  0  1  0  1  0  1

%e ... Reformatted by _Wolfdieter Lang_, May 12 2018

%e Inverse array begins

%e   n\k|  0   1   2   3   4   5   6   7

%e   ---+-------------------------------

%e    0 |  1

%e    1 | -1   1

%e    2 | -1   0   1

%e    3 |  1  -1  -1   1

%e    4 |  0   0  -1   0   1

%e    5 |  0   0   1  -1  -1   1

%e    6 |  0   0   0   0  -1   0   1

%e    7 |  0   0   0   0   1  -1  -1  1

%e   ... - _Peter Bala_, Aug 21 2021

%p T := (n, k) -> if igcd(n - k + 1, k + 1) mod 2 = 0 then 0 else 1 fi:

%p for n from 0 to 9 do seq(T(n, k), k = 0..n) od;

%p # Alternative:

%p T := (n, k) -> if n mod 2 = 1 then 1 else (k + 1) mod 2 fi:

%p for n from 0 to 9 do seq(T(n, k), k = 0..n) od; # _Peter Luschny_, Dec 12 2022

%t Table[Binomial[Mod[n, 2], Mod[k, 2]], {n, 0, 16}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Dec 12 2022 *)

%o (Python)

%o def A106465row(n: int) -> list[int]:

%o   if n % 2 == 1:

%o       return [1] * (n + 1)

%o   return [1, 0] * (n // 2) + [1]

%o for n in range(9): print(A106465row(n)) # _Peter Luschny_, Dec 12 2022

%Y Cf. A003989, A029578, A106466.

%K easy,nonn,tabl

%O 0,1

%A _Paul Barry_, May 03 2005

%E Edited and new name by _Peter Luschny_, Dec 12 2022