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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.

%I A048723 #16 Oct 26 2022 16:01:02
%S A048723 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,
%T A048723 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,
%U A048723 256,255,4096,1285,272,107,64,9,1
%N 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.
%H A048723 Alois P. Heinz, <a href="/A048723/b048723.txt">Antidiagonals n = 0..150, flattened</a>
%F A048723 a(n) = Xpower( (n-((trinv(n)*(trinv(n)-1))/2)), (((trinv(n)-1)*(((1/2)*trinv(n))+1))-n) );
%e A048723 1 0 0 0 0 0 0 0 0 ...
%e A048723 1 1 1 1 1 1 1 1 1 ...
%e A048723 1 2 4 8 16 32 64 128 256 ...
%e A048723 1 3 5 15 17 51 85 255 257 ...
%e A048723 1 4 16 64 256 1024 4096 16384 65536 ...
%p A048723 # Xmult and trinv have been given in A048720.
%p A048723 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;
%t A048723 trinv[n_] := Floor[(1 + Sqrt[1 + 8*n])/2];
%t A048723 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];
%t A048723 Xpower[nn_, mm_] := Xpower[nn, mm] = If[0 == mm, 1, Xmult[nn, Xpower[nn, mm - 1]]];
%t A048723 a[n_] := Xpower[n - (trinv[n]*(trinv[n] - 1))/2, (trinv[n] - 1)*((1/2)* trinv[n] + 1) - n];
%t A048723 Table[a[n], {n, 0, 65}] (* _Jean-François Alcover_, Mar 04 2016, adapted from Maple *)
%Y A048723 Cf. ordinary power table A004248 and A034369, A034373.
%Y A048723 Cf. A048710. Row 3: A001317, Row 5: A038183 (bisection of row 3), Row 7: A038184. Column 2: A000695, diagonal of A048720.
%Y A048723 Main diagonal: A048731.
%K A048723 nonn,tabl
%O A048723 0,9
%A A048723 _Antti Karttunen_, Apr 26 1999