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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. 139
 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 A. Karttunen, Scheme-program for computing this sequence. N. J. A. Sloane, Transforms: Maple implementation of binary eXclusive OR (XORnos). 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-François 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 September 21 11:45 EDT 2019. Contains 327253 sequences. (Running on oeis4.)