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A325825 Square array giving the monic polynomial q satisfying q = gcd(P(x),P(y)) where P(x) and P(y) are polynomials in ring GF(3)[X] with coefficients in {0,1,2} given by the ternary expansions of x and y. The polynomial q is converted back to a ternary number, and then expressed in decimal. 6
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 4, 5, 3, 5, 4, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 3, 4, 1, 3, 1, 1, 3, 1, 4, 3, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,13
COMMENTS
Array is symmetric, and is read by antidiagonals, with (x,y) = (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), ...
If there is a polynomial q that satisfies q = gcd(P(x),P(y)), then also polynomial -q (which is obtained by changing all nonzero coefficients of q as 1 <--> 2, see A004488) satisfies the same relation, because there are two units (+1 and -1) in polynomial ring GF(3)[X]. Here we always choose the polynomial that is monic (i.e., with a leading coefficient +1), thus its base-3 encoding has "1" as its most significant digit, and the terms given here are all included in A132141.
LINKS
EXAMPLE
The array begins as:
y
x 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...
--+-----------------------------------------------------
1 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
2 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
3 | 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, ...
4 | 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 4, 4, ...
5 | 1, 1, 1, 1, 5, 1, 5, 1, 1, 1, 5, 1, ...
6 | 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, ...
7 | 1, 1, 1, 1, 5, 1, 5, 1, 1, 1, 5, 1, ...
8 | 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 4, 4, ...
9 | 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, ...
10 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, ...
11 | 1, 1, 1, 4, 5, 1, 5, 4, 1, 1, 11, 4, ...
12 | 1, 1, 3, 4, 1, 3, 1, 4, 3, 1, 4, 12, ...
PROG
(PARI)
up_to = 105;
A004488(n) = subst(Pol(apply(x->(3-x)%3, digits(n, 3)), 'x), 'x, 3);
A325825sq(a, b) = { my(a=fromdigits(Vec(lift(gcd(Pol(digits(a, 3))*Mod(1, 3), Pol(digits(b, 3))*Mod(1, 3)))), 3), b=A004488(a)); min(a, b); };
A325825list(up_to) = { my(v = vector(up_to), i=0); for(a=1, oo, for(col=1, a, i++; if(i > up_to, return(v)); v[i] = A325825sq(col, (a-(col-1))))); (v); };
v325825 = A325825list(up_to);
A325825(n) = v325825[n];
CROSSREFS
Central diagonal: A330740 (after its initial zero).
Sequence in context: A327537 A120263 A250208 * A030580 A030579 A030578
KEYWORD
nonn,tabl
AUTHOR
Antti Karttunen, May 22 2019
STATUS
approved

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Last modified February 25 13:12 EST 2024. Contains 370330 sequences. (Running on oeis4.)