A048723 Binary "exponentiation" without carries: {0..y}^{0..x}, where y (column index) is binary encoding of GF(2)-polynomial and x (row index) is the exponent.

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 3, 1, 0, 1, 8, 5, 4, 1, 0, 1, 16, 15, 16, 5, 1, 0, 1, 32, 17, 64, 17, 6, 1, 0, 1, 64, 51, 256, 85, 20, 7, 1, 0, 1, 128, 85, 1024, 257, 120, 21, 8, 1, 0, 1, 256, 255, 4096, 1285, 272, 107, 64, 9, 1

Offset

0,9

Links

Alois P. Heinz, Antidiagonals n = 0..150, flattened

Formula

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

Example

1 0 0 0 0 0 0 0 0 ...
1 1 1 1 1 1 1 1 1 ...
1 2 4 8 16 32 64 128 256 ...
1 3 5 15 17 51 85 255 257 ...
1 4 16 64 256 1024 4096 16384 65536 ...

Maple

# Xmult and trinv have been given in A048720.
Xpower := proc(nn, mm) option remember; if(0 = mm) then RETURN(1); # By definition, also 0^0 = 1. else RETURN(Xmult(nn, Xpower(nn, mm-1))); fi; 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]; s];
Xpower[nn_, mm_] := Xpower[nn, mm] = If[0 == mm, 1, Xmult[nn, Xpower[nn, mm - 1]]];
a[n_] := Xpower[n - (trinv[n]*(trinv[n] - 1))/2, (trinv[n] - 1)*((1/2)* trinv[n] + 1) - n];
Table[a[n], {n, 0, 65}] (* Jean-François Alcover, Mar 04 2016, adapted from Maple *)

Crossrefs

Cf. ordinary power table A004248 and A034369, A034373.

Cf. A048710. Row 3: A001317, Row 5: A038183 (bisection of row 3), Row 7: A038184. Column 2: A000695, diagonal of A048720.

Main diagonal: A048731.

Sequence in context: A055340 A119328 A058716 * A364386 A088455 A361390

Adjacent sequences: A048720 A048721 A048722 * A048724 A048725 A048726

Keyword

nonn,tabl

Author

Antti Karttunen, Apr 26 1999

Status

approved