A325825 - OEIS

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.

%I #41 Jan 13 2020 20:39:14

%S 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,

%T 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,

%U 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

%N 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.

%C Array is symmetric, and is read by antidiagonals, with (x,y) = (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), ...

%C 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.

%e The array begins as:

%e y

%e x 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...

%e --+-----------------------------------------------------

%e 1 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 2 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 3 | 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, ...

%e 4 | 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 4, 4, ...

%e 5 | 1, 1, 1, 1, 5, 1, 5, 1, 1, 1, 5, 1, ...

%e 6 | 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, ...

%e 7 | 1, 1, 1, 1, 5, 1, 5, 1, 1, 1, 5, 1, ...

%e 8 | 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 4, 4, ...

%e 9 | 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, ...

%e 10 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, ...

%e 11 | 1, 1, 1, 4, 5, 1, 5, 4, 1, 1, 11, 4, ...

%e 12 | 1, 1, 3, 4, 1, 3, 1, 4, 3, 1, 4, 12, ...

%o (PARI)

%o up_to = 105;

%o A004488(n) = subst(Pol(apply(x->(3-x)%3, digits(n, 3)), 'x), 'x, 3);

%o 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); };

%o 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); };

%o v325825 = A325825list(up_to);

%o A325825(n) = v325825[n];

%Y Cf. A004488, A091255, A132141, A325820, A325821, A325827.

%Y Central diagonal: A330740 (after its initial zero).

%K nonn,tabl

%O 1,13

%A _Antti Karttunen_, May 22 2019