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Search: id:A003987
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| A003987 |
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Table of n XOR m (or Nim-sum of n and m) read by antidiagonals, i.e. with entries in the order (n,m) = (0,0),(0,1),(1,0),(0,2),(1,1),(2,0),... |
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43
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| 0, 1, 1, 2, 0, 2, 3, 3, 3, 3, 4, 2, 0, 2, 4, 5, 5, 1, 1, 5, 5, 6, 4, 6, 0, 6, 4, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 6, 4, 6, 0, 6, 4, 6, 8, 9, 9, 5, 5, 1, 1, 5, 5, 9, 9, 10, 8, 10, 4, 2, 0, 2, 4, 10, 8, 10, 11, 11, 11, 11, 3, 3, 3, 3, 11, 11, 11, 11, 12, 10, 8, 10, 12, 2, 0, 2, 12, 10, 8, 10, 12, 13, 13, 9, 9
(list; table; graph; listen)
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OFFSET
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0,4
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COMMENT
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T[2i,2j] = 2T[i,j], T[2i+1,2j] = 2T[i,j] + 1.
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REFERENCES
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J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.
J. H. Conway, On Numbers and Games. Academic Press, NY, 1976, pp. 51-53.
D. Gale, Tracking the Automatic Ant and Other Mathematical Explorations, A Collection of Mathematical Entertainments Columns from The Mathematical Intelligencer, Springer, 1998; see p. 190. [From N. J. A. Sloane, Jul 14 2009]
R. K. Guy, Impartial games, pp. 35-55 of Combinatorial Games, ed. R. K. Guy, Proc. Sympos. Appl. Math., 43, Amer. Math. Soc., 1991.
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LINKS
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T. D. Noe, Rows n=0..100 of triangle, flattened
J.-P. Allouche and J. Shallit, The Ring of k-regular Sequences, II
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
Index entries for sequences related to Nim-sums
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EXAMPLE
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Table begins
0 1 2 3 4 5 6 7 ...
1 0 3 2 5 4 7 6 ...
2 3 0 1 6 7 4 5 ...
3 2 1 0 7 6 5 3 ...
4 5 6 7 0 1 2 3 ...
...................
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MAPLE
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nimsum := proc(a, b) local t1, t2, t3, t4, l; t1 := convert(a+2^20, base, 2); t2 := convert(b+2^20, base, 2); t3 := evalm(t1+t2); map(x->x mod 2, t3); t4 := convert(evalm(%), list); l := convert(t4, base, 2, 10); sum(l[k]*10^(k-1), k=1..nops(l)); end; # memo: adjust 2^20 to be much bigger than a and b
AT := array(0..N, 0..N); for a from 0 to N do for b from a to N do AT[a, b] := nimsum(a, b); AT[b, a] := AT[a, b]; od: od:
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MATHEMATICA
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Flatten[Table[BitXor[b, a - b], {a, 0, 10}, {b, 0, a}]] (BitXor and Nim Sum are equivalent)
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CROSSREFS
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Initial rows are A001477, A004442, A004443, A004444, etc. Cf. A051775, A051776.
Cf. A003986 (OR) and A004198 (AND).
Antidiagonal sums are in A006582.
Sequence in context: A074660 A002125 A135356 this_sequence A141692 A063180 A141693
Adjacent sequences: A003984 A003985 A003986 this_sequence A003988 A003989 A003990
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KEYWORD
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tabl,nonn,nice
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AUTHOR
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Marc LeBrun (mlb(AT)well.com)
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