This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A106465 #55 Mar 21 2023 15:32:21
%S A106465 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 A106465 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 A106465 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 A106465 Triangle read by rows, T(n, k) = 1 if n mod 2 = 1, otherwise (k + 1) mod 2.
%C A106465 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 A106465 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 A106465 Row sums are A029578(n+2). Antidiagonal sums are A106466.
%C A106465 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 A106465 From _Peter Bala_, Aug 21 2021: (Start)
%C A106465 Using the notation of Davenport et al.:
%C A106465 This is the double Riordan array ( 1/(1 - x); x/(1 + x), x*(1 + x) ).
%C A106465 The inverse array equals ( (1 - x)*(1 - x^2); x*(1 - x), x*(1 + x) ).
%C A106465 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 A106465 Peter Bala, <a href="/A177994/a177994.pdf">Matrices with repeated columns - the generalised Appell groups</a>
%H A106465 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 A106465 If gcd(n - k + 1, k + 1) mod 2 = 0 then T(n, k) = 0, otherwise T(n, k) = 1.
%F A106465 T(n, k) = A003989(n + 1, k + 1) mod 2.
%F A106465 T(n, k) = binomial(n mod 2, k mod 2). - _Peter Luschny_, Dec 12 2022
%e A106465 The triangle begins:
%e A106465 n\k| 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...
%e A106465 ---+------------------------------------------------
%e A106465 0 | 1
%e A106465 1 | 1 1
%e A106465 2 | 1 0 1
%e A106465 3 | 1 1 1 1
%e A106465 4 | 1 0 1 0 1
%e A106465 5 | 1 1 1 1 1 1
%e A106465 6 | 1 0 1 0 1 0 1
%e A106465 7 | 1 1 1 1 1 1 1 1
%e A106465 8 | 1 0 1 0 1 0 1 0 1
%e A106465 9 | 1 1 1 1 1 1 1 1 1 1
%e A106465 10 | 1 0 1 0 1 0 1 0 1 0 1
%e A106465 11 | 1 1 1 1 1 1 1 1 1 1 1 1
%e A106465 12 | 1 0 1 0 1 0 1 0 1 0 1 0 1
%e A106465 13 | 1 1 1 1 1 1 1 1 1 1 1 1 1 1
%e A106465 14 | 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
%e A106465 ... Reformatted by _Wolfdieter Lang_, May 12 2018
%e A106465 Inverse array begins
%e A106465 n\k| 0 1 2 3 4 5 6 7
%e A106465 ---+-------------------------------
%e A106465 0 | 1
%e A106465 1 | -1 1
%e A106465 2 | -1 0 1
%e A106465 3 | 1 -1 -1 1
%e A106465 4 | 0 0 -1 0 1
%e A106465 5 | 0 0 1 -1 -1 1
%e A106465 6 | 0 0 0 0 -1 0 1
%e A106465 7 | 0 0 0 0 1 -1 -1 1
%e A106465 ... - _Peter Bala_, Aug 21 2021
%p A106465 T := (n, k) -> if igcd(n - k + 1, k + 1) mod 2 = 0 then 0 else 1 fi:
%p A106465 for n from 0 to 9 do seq(T(n, k), k = 0..n) od;
%p A106465 # Alternative:
%p A106465 T := (n, k) -> if n mod 2 = 1 then 1 else (k + 1) mod 2 fi:
%p A106465 for n from 0 to 9 do seq(T(n, k), k = 0..n) od; # _Peter Luschny_, Dec 12 2022
%t A106465 Table[Binomial[Mod[n, 2], Mod[k, 2]], {n, 0, 16}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Dec 12 2022 *)
%o A106465 (Python)
%o A106465 def A106465row(n: int) -> list[int]:
%o A106465 if n % 2 == 1:
%o A106465 return [1] * (n + 1)
%o A106465 return [1, 0] * (n // 2) + [1]
%o A106465 for n in range(9): print(A106465row(n)) # _Peter Luschny_, Dec 12 2022
%Y A106465 Cf. A003989, A029578, A106466.
%K A106465 easy,nonn,tabl
%O A106465 0,1
%A A106465 _Paul Barry_, May 03 2005
%E A106465 Edited and new name by _Peter Luschny_, Dec 12 2022