A034369 -id:A034369

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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.

+10
22

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

(list; table; graph; refs; listen; history; text; internal format)

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.

KEYWORD

nonn,tabl

AUTHOR

Antti Karttunen, Apr 26 1999

STATUS

approved

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