OFFSET
1,5
COMMENTS
Array is read by antidiagonals, with (x,y) = (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), ...
Analogous to A003989.
"Coded in binary" means that a polynomial a(n)*X^n+...+a(0)*X^0 over GF(2) is represented by the binary number a(n)*2^n+...+a(0)*2^0 in Z (where a(k)=0 or 1).
LINKS
FORMULA
A(x,y) = A(y,x) = A(x, A003987(x,y)) = A(A003987(x,y), y), where A003987 gives the bitwise-XOR of its two arguments. - Antti Karttunen, Sep 28 2019
EXAMPLE
The top left 17 X 17 corner of the array:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17
+---------------------------------------------------------------
1: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
2: 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, ...
3: 1, 1, 3, 1, 3, 3, 1, 1, 3, 3, 1, 3, 1, 1, 3, 1, 3, ...
4: 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, ...
5: 1, 1, 3, 1, 5, 3, 1, 1, 3, 5, 1, 3, 1, 1, 5, 1, 5, ...
6: 1, 2, 3, 2, 3, 6, 1, 2, 3, 6, 1, 6, 1, 2, 3, 2, 3, ...
7: 1, 1, 1, 1, 1, 1, 7, 1, 7, 1, 1, 1, 1, 7, 1, 1, 1, ...
8: 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 8, 1, ...
9: 1, 1, 3, 1, 3, 3, 7, 1, 9, 3, 1, 3, 1, 7, 3, 1, 3, ...
10: 1, 2, 3, 2, 5, 6, 1, 2, 3, 10, 1, 6, 1, 2, 5, 2, 5, ...
11: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, ...
12: 1, 2, 3, 4, 3, 6, 1, 4, 3, 6, 1, 12, 1, 2, 3, 4, 3, ...
13: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, ...
14: 1, 2, 1, 2, 1, 2, 7, 2, 7, 2, 1, 2, 1, 14, 1, 2, 1, ...
15: 1, 1, 3, 1, 5, 3, 1, 1, 3, 5, 1, 3, 1, 1, 15, 1, 15, ...
16: 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 16, 1, ...
17: 1, 1, 3, 1, 5, 3, 1, 1, 3, 5, 1, 3, 1, 1, 15, 1, 17, ...
...
3, which is "11" in binary, encodes polynomial X + 1, while 7 ("111" in binary) encodes polynomial X^2 + X + 1, whereas 9 ("1001" in binary), encodes polynomial X^3 + 1. Now (X + 1)(X^2 + X + 1) = (X^3 + 1) when the polynomials are multiplied over GF(2), or equally, when multiplication of integers 3 and 7 is done as a carryless base-2 product (A048720(3,7) = 9). Thus it follows that A(3,9) = A(9,3) = 3 and A(7,9) = A(9,7) = 7.
Furthermore, 5 ("101" in binary) encodes polynomial X^2 + 1 which is equal to (X + 1)(X + 1) in GF(2)[X], thus A(5,9) = A(9,5) = 3, as the irreducible polynomial (X + 1) is the only common factor for polynomials X^2 + 1 and X^3 + 1.
PROG
(PARI) A091255sq(a, b) = fromdigits(Vec(lift(gcd(Pol(binary(a))*Mod(1, 2), Pol(binary(b))*Mod(1, 2)))), 2); \\ Antti Karttunen, Aug 12 2019
CROSSREFS
AUTHOR
Antti Karttunen, Jan 03 2004
EXTENSIONS
Data section extended up to a(105), examples added by Antti Karttunen, Sep 28 2019
STATUS
approved