A054237 -id:A054237

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A329331

Binary operation over the nonnegative integers, distributive over A003987(.,.), such that A(2^i, 2^j) = 2^A054237(i,j). Square array A(n,k), n >= 0, k >= 0, read by descending antidiagonals.

+20
1

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 8, 3, 0, 0, 4, 10, 10, 4, 0, 0, 5, 16, 9, 16, 5, 0, 0, 6, 18, 20, 20, 18, 6, 0, 0, 7, 24, 23, 32, 23, 24, 7, 0, 0, 8, 26, 30, 36, 36, 30, 26, 8, 0, 0, 9, 64, 29, 48, 33, 48, 29, 64, 9, 0, 0, 10, 66, 72, 52, 54, 54, 52, 72, 66, 10, 0, 0, 11, 72, 75, 128, 51, 40, 51, 128, 75, 72, 11, 0

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OFFSET

0,8

COMMENTS

This sequence defines a multiplication operation that goes with bitwise exclusive-or (A003987) as addition operation to form a ring over the nonnegative integers. It is isomorphic to the polynomial ring GF(2)[x,y], as is the ring defined in A329329.

The ring defined by A329329 is unusual in that it has A059897(.,.) as its addition operation, given that A059897 has more similarities to integer multiplication. A003987, which is isomorphic to A059897 as a binary operation, seems a more standard choice for an addition operator.

However, as explained in A329329, A059897 has a natural choice for mapping a generating set to the 2-dimensions (x and y) of the generating set for the additive group of GF(2)[x,y]. Instead, A003987 needs a pairing function to map its most natural generating set {2^k: k >= 0} onto {x^i * y^j: i >= 0, j >= 0}.

The choice made here was to map 2^k onto the 2 dimensions of x^i * y^j, by proceeding through x and y dimensions as when reading an array by antidiagonals. 2^0 = 1 is mapped to (x^0 * y*0) = 1, 2^1 = 2 is mapped to (x^1 * y^0) = x, 2^2 = 4 to (x^0 * y^1) = y, 8 to (x^2 * y^0) = x^2, and so on, 16 mapped to xy, 32 to y^2, 64 to x^3, etc. With this mapping, it can be shown that the result of the multiplying the polynomial images of 2^i and 2^j is the image of 2^A054237(i,j).

LINKS

Table of n, a(n) for n=0..90.

Eric Weisstein's World of Mathematics, Distributive

Eric Weisstein's World of Mathematics, Pairing Function

Eric Weisstein's World of Mathematics, Ring

Wikipedia, Generating set of a group

Wikipedia, Polynomial ring

FORMULA

A(2^i, 2^j) = 2^A054237(i,j).

A(A003987(n,m), k) = A003987(A(n,k), A(m,k)).

A(n, A003987(m,k)) = A003987(A(n,m), A(n,k)).

Derived formulas:(Start)

A(n,k) = A(k,n).

A(n,0) = A(0,k) = 0.

A(n,1) = A(1,n) = n.

A(n, A(m,k)) = A(A(n,m), k).

(End)

EXAMPLE

Square array A(n,k) begins:

n\k | 0 1 2 3 4 5 6 7 8 9 10

----+----------------------------------------------------------------

0 | 0 0 0 0 0 0 0 0 0 0 0

1 | 0 1 2 3 4 5 6 7 8 9 10

2 | 0 2 8 10 16 18 24 26 64 66 72

3 | 0 3 10 9 20 23 30 29 72 75 66

4 | 0 4 16 20 32 36 48 52 128 132 144

5 | 0 5 18 23 36 33 54 51 136 141 154

6 | 0 6 24 30 48 54 40 46 192 198 216

7 | 0 7 26 29 52 51 46 41 200 207 210

8 | 0 8 64 72 128 136 192 200 1024 1032 1088

9 | 0 9 66 75 132 141 198 207 1032 1025 1098

10 | 0 10 72 66 144 154 216 210 1088 1098 1032

CROSSREFS

Cf. A003987, A054237, A059897, A329329.

KEYWORD

nonn,tabl

AUTHOR

Peter Munn, Nov 10 2019

STATUS

approved

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