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
Examples
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 ...
Links
- Alois P. Heinz, Antidiagonals n = 0..150, flattened
Crossrefs
Programs
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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;
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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 *)
Formula
a(n) = Xpower( (n-((trinv(n)*(trinv(n)-1))/2)), (((trinv(n)-1)*(((1/2)*trinv(n))+1))-n) );