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A301636
Square array T(n, k) read by antidiagonals upwards, n >= 0 and k >= 0: T(n, k) = square of the distance from n + k*i to nearest square of a Gaussian integer (where i denotes the root of -1 with positive imaginary part).
2
0, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 2, 4, 2, 4, 1, 1, 4, 2, 4, 9, 4, 2, 4, 1, 1, 5, 4, 4, 5, 5, 2, 0, 2, 5, 1, 1, 5, 8, 5, 1, 1, 5, 2, 0, 0, 2, 8, 10, 4, 2, 4, 5, 1, 1, 1, 1, 5, 10, 8, 5, 5, 9, 4, 2, 4, 4, 2, 4, 9, 5, 5, 8, 10, 9, 5, 5, 9, 9, 5, 5, 9, 4, 2, 4, 10
OFFSET
0,8
COMMENTS
The distance between two Gaussian integers is not necessarily integer, hence the use of the square of the distance.
This sequence is a complex variant of A053188.
See A301626 for the square array dealing with cubes of Gaussian integers.
FORMULA
T(n, 0) <= A053188(n)^2.
T(n, 0) = 0 iff n is a square (A000290).
T(0, k) = 0 iff k is twice a square (A001105).
T(n, k) = 0 iff n + k*i = z^2 for some Gaussian integer z.
EXAMPLE
Square array begins:
n\k| 0 1 2 3 4 5 6 7 8 9 10
---+-------------------------------------------------------
0| 0 1 0 1 4 9 4 1 0 1 4
1| 0 1 1 2 4 5 5 2 1 2 5
2| 1 2 4 2 1 2 5 5 4 5 8
3| 1 2 4 1 0 1 4 9 9 10 8
4| 0 1 4 2 1 2 5 10 16 10 5
5| 1 2 5 5 4 5 8 10 13 9 4
6| 4 5 8 10 8 5 4 5 8 10 5
7| 4 5 8 10 5 2 1 2 5 10 8
8| 1 2 5 9 4 1 0 1 4 9 13
9| 0 1 4 9 5 2 1 2 5 10 17
10| 1 2 5 10 8 5 4 5 8 13 20
PROG
(PARI) See Links section.
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Rémy Sigrist, Mar 25 2018
STATUS
approved