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A048720 Multiplication table {0..i} X {0..j} of binary polynomials (in GF(2)) interpreted as binary vectors, then written in base 10; or, binary multiplication without carries. 115
0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 4, 3, 0, 0, 4, 6, 6, 4, 0, 0, 5, 8, 5, 8, 5, 0, 0, 6, 10, 12, 12, 10, 6, 0, 0, 7, 12, 15, 16, 15, 12, 7, 0, 0, 8, 14, 10, 20, 20, 10, 14, 8, 0, 0, 9, 16, 9, 24, 17, 24, 9, 16, 9, 0, 0, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 0, 0, 11, 20, 27, 32, 27, 20, 27, 32, 27, 20, 11, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

Essentially same as A091257 but computed starting from offset 0 instead of 1.

LINKS

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

A. Karttunen, Scheme-program for computing this sequence.

N. J. A. Sloane, Transforms: Maple implementation of binary eXclusive OR (XORnos).

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

FORMULA

a(n) = Xmult( (((trinv(n)-1)*(((1/2)*trinv(n))+1))-n), (n-((trinv(n)*(trinv(n)-1))/2)) );

T(2b, c)=T(c, 2b)=T(b, 2c)=2T(b, c); T(2b+1, c)=T(c, 2b+1)=2T(b, c) XOR c - Henry Bottomley, Mar 16 2001

EXAMPLE

Top left corner of array:

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ...

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 ...

0 3 6 5 12 15 10 9 24 27 30 29 20 23 18 17 ...

MAPLE

trinv := n -> floor((1+sqrt(1+8*n))/2); # Gives integral inverses of the triangular numbers

# Binary multiplication of nn and mm, but without carries (use XOR instead of ADD):

Xmult := proc(nn, mm) local n, m, s; n := nn; m := mm; s := 0; while (n > 0) do if(1 = (n mod 2)) then s := XORnos(s, m); fi; n := floor(n/2); # Shift n right one bit. m := m*2; # Shift m left one bit. od; RETURN(s); end;

MATHEMATICA

trinv[n_] := Floor[(1 + Sqrt[1 + 8*n])/2];

Xmult[nn_, mm_] := Module[{n = nn, m = mm, s = 0}, While[n > 0, If[1 == Mod[n, 2], s = BitXor[s, m]]; n = Floor[n/2]; m = m*2]; Return[s]];

a[n_] := Xmult[(trinv[n] - 1)*((1/2)*trinv[n] + 1) - n, n - (trinv[n]*(trinv[n] - 1))/2];

Table[a[n], {n, 0, 100}] (* Jean-Fran├žois Alcover, Mar 16 2015, updated Mar 06 2016 after Maple *)

CROSSREFS

Cf. A048631, A051776, A059692.

Ordinary {0..i} * {0..j} multiplication table: A004247 and its differences from this: A061858.

Binary irreducible polynomials ("X-primes"): A014580, table of "X-powers": A048723. Row/column 3: A048724, 5: A048725, 6: A048726, 7: A048727.

Sequence in context: A263139 A063711 A057893 * A067138 A059692 A004247

Adjacent sequences:  A048717 A048718 A048719 * A048721 A048722 A048723

KEYWORD

nonn,tabl

AUTHOR

Antti Karttunen, Apr 26 1999

STATUS

approved

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Last modified January 20 17:24 EST 2018. Contains 297960 sequences.