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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

Table of n, a(n) for n=1..105.

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

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

Central diagonal: A330740 (after its initial zero).

Sequence in context: A327537 A120263 A250208 * A030580 A030579 A030578

Adjacent sequences:  A325822 A325823 A325824 * A325826 A325827 A325828

KEYWORD

nonn,tabl

AUTHOR

Antti Karttunen, May 22 2019

STATUS

approved

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Last modified July 24 23:25 EDT 2021. Contains 346273 sequences. (Running on oeis4.)