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Search: id:A048720
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| A048720 |
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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. |
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+0
59
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| 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; listen)
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OFFSET
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0,8
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COMMENT
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Essentially same as A091257 but computed starting from offset 0 instead of 1.
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LINKS
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N. J. A. Sloane, Maple implementation of binary eXclusive OR (XORnos)
A. Karttunen, Scheme-program for computing this sequence.
Index entries for sequences operating on GF(2)[X]-polynomials
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FORMULA
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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 (se16(AT)btinternet.com), Mar 16 2001
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EXAMPLE
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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 ...
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MAPLE
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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;
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CROSSREFS
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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: A108036 A063711 A057893 this_sequence A067138 A059692 A004247
Adjacent sequences: A048717 A048718 A048719 this_sequence A048721 A048722 A048723
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KEYWORD
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nonn,tabl
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AUTHOR
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Antti Karttunen, Apr 26 1999
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