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A091255 Square array computed from gcd(P(x),P(y)) where P(x) and P(y) are polynomials with coefficients in {0,1} given by the binary expansions of x and y, and the polynomial calculation is done over GF(2), with the result converted back to a binary number, and then expressed in decimal. Array is symmetric, and is read by falling antidiagonals. 33
1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 3, 2, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 2, 1, 2, 5, 2, 1, 2, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 3, 2, 7, 2, 3, 2, 1, 2, 1, 1, 1, 3, 1, 5, 3, 1, 1, 3, 5, 1, 3, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
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
Cf. also A327856 (the upper left triangular section of this array), A327857.
Sequence in context: A159923 A287957 A003989 * A332013 A324350 A175466
KEYWORD
nonn,tabl,look
AUTHOR
Antti Karttunen, Jan 03 2004
EXTENSIONS
Data section extended up to a(105), examples added by Antti Karttunen, Sep 28 2019
STATUS
approved

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Last modified March 19 06:17 EDT 2024. Contains 370952 sequences. (Running on oeis4.)