A106449 Square array (P(x) XOR P(y))/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 all calculations are done in polynomial ring GF(2)[X], with the result converted back to a binary number, and then expressed in decimal. Array is symmetric, and is read by antidiagonals.

0, 3, 3, 2, 0, 2, 5, 1, 1, 5, 4, 3, 0, 3, 4, 7, 7, 7, 7, 7, 7, 6, 2, 2, 0, 2, 2, 6, 9, 5, 3, 1, 1, 3, 5, 9, 8, 5, 4, 1, 0, 1, 4, 5, 8, 11, 11, 11, 3, 1, 1, 3, 11, 11, 11, 10, 4, 6, 3, 2, 0, 2, 3, 6, 4, 10, 13, 9, 7, 13, 13, 1, 1, 13, 13, 7, 9, 13, 12, 7, 8, 7, 4, 7, 0, 7, 4, 7, 8, 7, 12, 15, 15, 5

Offset

1,2

Comments

Array is read by antidiagonals, with row x and column y ranging as: (x,y) = (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), ...

"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).

This is GF(2)[X] analog of A106448. In the definition XOR means addition in polynomial ring GF(2)[X], that is, a carryless binary addition, A003987.

Links

Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of array

Index entries for sequences operating on GF(2)[X]-polynomials

Formula

A(x, y) = A280500(A003987(x, y), A091255(x, y)), that is, A003987(x, y) = A048720(A(x, y), A091255(x, y)).

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 :  0,  3,  2,  5,  4,  7,  6,  9,  8, 11, 10, 13, 12, 15, 14, 17, 16, ...
   2 :  3,  0,  1,  3,  7,  2,  5,  5, 11,  4,  9,  7, 15,  6, 13,  9, 19, ...
   3 :  2,  1,  0,  7,  2,  3,  4, 11,  6,  7,  8,  5, 14, 13,  4, 19, 14, ...
   4 :  5,  3,  7,  0,  1,  1,  3,  3, 13,  7, 15,  2,  9,  5, 11,  5, 21, ...
   5 :  4,  7,  2,  1,  0,  1,  2, 13,  4,  3, 14,  7,  8, 11,  2, 21,  4, ...
   6 :  7,  2,  3,  1,  1,  0,  1,  7,  5,  2, 13,  3, 11,  4,  7, 11, 13, ...
   7 :  6,  5,  4,  3,  2,  1,  0, 15,  2, 13, 12, 11, 10,  3,  8, 23, 22, ...
   8 :  9,  5, 11,  3, 13,  7, 15,  0,  1,  1,  3,  1,  5,  3,  7,  3, 25, ...
   9 :  8, 11,  6, 13,  4,  5,  2,  1,  0,  1,  2,  3,  4,  1,  2, 25,  8, ...
  10 : 11,  4,  7,  7,  3,  2, 13,  1,  1,  0,  1,  1,  7,  2,  1, 13,  7, ...
  11 : 10,  9,  8, 15, 14, 13, 12,  3,  2,  1,  0,  7,  6,  5,  4, 27, 26, ...
  12 : 13,  7,  5,  2,  7,  3, 11,  1,  3,  1,  7,  0,  1,  1,  1,  7, 11, ...
  13 : 12, 15, 14,  9,  8, 11, 10,  5,  4,  7,  6,  1,  0,  3,  2, 29, 28, ...
  14 : 15,  6, 13,  5, 11,  4,  3,  3,  1,  2,  5,  1,  3,  0,  1, 15, 31, ...
  15 : 14, 13,  4, 11,  2,  7,  8,  7,  2,  1,  4,  1,  2,  1,  0, 31,  2, ...
  16 : 17,  9, 19,  5  21, 11, 23,  3, 25, 13, 27,  7, 29, 15, 31,  0,  1, ...
  17 : 16, 19, 14, 21,  4, 13, 22, 25,  8,  7, 26, 11, 28, 31,  2,  1,  0, ...

Prog

(PARI)
up_to = 105;
A106449sq(a, b) = { my(Pa=Pol(binary(a))*Mod(1, 2), Pb=Pol(binary(b))*Mod(1, 2)); fromdigits(Vec(lift((Pa+Pb)/gcd(Pa, Pb))), 2); }; \\ Note that XOR is just + in GF(2)[X] world.
A106449list(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] = A106449sq(col, (a-(col-1))))); (v); };
v106449 = A106449list(up_to);
A106449(n) = v106449[n]; \\ Antti Karttunen, Oct 21 2019

Crossrefs

Row 1: A004442 (without its initial term), row 2: A106450 (without its initial term).

Cf. A003987, A048720, A091255, A106448, A280500.

Sequence in context: A181407 A114187 A016037 * A256522 A097278 A254411

Adjacent sequences: A106446 A106447 A106448 * A106450 A106451 A106452

Keyword

nonn,tabl

Author

Antti Karttunen, May 21 2005

Status

approved