|
|
A051933
|
|
Triangle T(n,m) = Nim-sum (or XOR) of n and m, read by rows, 0<=m<=n.
|
|
6
|
|
|
0, 1, 0, 2, 3, 0, 3, 2, 1, 0, 4, 5, 6, 7, 0, 5, 4, 7, 6, 1, 0, 6, 7, 4, 5, 2, 3, 0, 7, 6, 5, 4, 3, 2, 1, 0, 8, 9, 10, 11, 12, 13, 14, 15, 0, 9, 8, 11, 10, 13, 12, 15, 14, 1, 0, 10, 11, 8, 9, 14, 15, 12, 13, 2, 3, 0, 11, 10, 9, 8, 15, 14, 13, 12, 3, 2, 1, 0, 12, 13, 14, 15, 8, 9, 10, 11, 4, 5, 6, 7, 0
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,4
|
|
REFERENCES
|
E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 60.
J. H. Conway, On Numbers and Games, Academic Press, p. 52.
|
|
LINKS
|
|
|
EXAMPLE
|
{0},
{1,0},
{2,3,0},
{3,2,1,0}, ...
|
|
MAPLE
|
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:
# Alternative:
A051933 := (n, k) -> Bits:-Xor(n, k):
|
|
MATHEMATICA
|
|
|
PROG
|
(Haskell)
import Data.Bits (xor)
a051933 n k = n `xor` k :: Int
a051933_row n = map (a051933 n) [0..n]
a051933_tabl = map a051933_row [0..]
(Julia)
using IntegerSequences
A051933Row(n) = [Bits("XOR", n, k) for k in 0:n]
for n in 0:10 println(A051933Row(n)) end # Peter Luschny, Sep 25 2021
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
More terms from Michael Lugo (mlugo(AT)thelabelguy.com), Dec 22 1999
|
|
STATUS
|
approved
|
|
|
|