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Search: id:A118666
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| A118666 |
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Words that, as binary polynomials p(x) over GF(2), are fixed points of the map p(x) |--> p(x+1). |
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1
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| 0, 1, 6, 7, 18, 19, 20, 21, 106, 107, 108, 109, 120, 121, 126, 127, 258, 259, 260, 261, 272, 273, 278, 279, 360, 361, 366, 367, 378, 379, 380, 381, 1546, 1547, 1548, 1549, 1560, 1561, 1566, 1567, 1632, 1633, 1638, 1639, 1650, 1651, 1652, 1653, 1800, 1801
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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If p(x) is a fixed point then P(x):=(x+x^2)*p(x) and P(x)+1 are also fixed points.
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LINKS
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Joerg Arndt fxtbook, section "Invertible transforms on words" in chapter "Bit wizardry"
Index entries for sequences operating on (or containing) GF(2)[X]-polynomials
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EXAMPLE
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a(4)=8 corresponds to the polynomial p(x)=x^4+x (18 is 10010 in binary).
p(x+1) = (x+1)^4 + (x+1) = x^4 + 4*x^3 + 6*x^2 + 5*x + 2 = x^4+x = p(x)
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PROGRAM
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// C++ function that returns a unique fixed point for each argument: ulong A(ulong s) { if ( 0==s ) return 0; ulong f = 1; while ( s>1 ) { f ^= (f<<1); f <<= 1; f |= (s&1); s >>= 1; } return f; } // the elements are not produced in increasing order, but as follows // 0 1 6 7 20 18 21 19 120 108 126 106 121 109 127 107 272 360 ...
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CROSSREFS
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Sequence in context: A008538 A000870 A062850 this_sequence A030746 A005302 A028324
Adjacent sequences: A118663 A118664 A118665 this_sequence A118667 A118668 A118669
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KEYWORD
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base,nonn
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AUTHOR
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Joerg Arndt (arndt(AT)jjj.de), May 19 2006, May 20 2006
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