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A106465
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Triangle read by rows, T(n, k) = 1 if n mod 2 = 1, otherwise (k + 1) mod 2.
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6
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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
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OFFSET
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0,1
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COMMENTS
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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).
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
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)
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LINKS
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D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012).
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FORMULA
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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
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EXAMPLE
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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
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
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MAPLE
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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
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MATHEMATICA
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Table[Binomial[Mod[n, 2], Mod[k, 2]], {n, 0, 16}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 12 2022 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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