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A163355
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Permutation of integers for constructing Hilbert curve in N x N grid.
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24
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0, 1, 3, 2, 14, 15, 13, 12, 4, 7, 5, 6, 8, 11, 9, 10, 16, 19, 17, 18, 20, 21, 23, 22, 30, 29, 31, 28, 24, 25, 27, 26, 58, 57, 59, 56, 54, 53, 55, 52, 60, 61, 63, 62, 50, 51, 49, 48, 32, 35, 33, 34, 36, 37, 39, 38, 46, 45, 47, 44, 40, 41, 43, 42, 234, 235, 233, 232, 236, 239
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OFFSET
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0,3
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LINKS
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FORMULA
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a(0) = 0,
and given d=1, 2 or 3, then a((d*(4^i))+r)
= (4^i) + a(A057300(r)), if d=1 and i is even, or if d=2 and i is odd
= 3*(4^i) + a((4^i)-1-r) in other cases.
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MAPLE
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option remember;
`if`(n=0, 0, procname(iquo(n, 4, 'r'))*4+[0, 2, 1, 3][r+1])
end proc:
option remember ;
local d, base4, i, r ;
if n <= 1 then
return n ;
end if;
base4 := convert(n, base, 4) ;
d := op(-1, base4) ;
i := nops(base4)-1 ;
r := n-d*4^i ;
if ( d=1 and type(i, even) ) or ( d=2 and type(i, odd)) then
elif d= 3 then
else
3*4^i+procname(4^i-1-r) ;
end if;
end proc:
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PROG
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(MIT Scheme:) (define (A163355 n) (let* ((i (floor->exact (/ (A000523 n) 2))) (dd (modulo (floor->exact (/ n (expt 4 i))) 4)) (r (if (zero? n) n (modulo n (expt 4 i))))) (cond ((zero? n) n) ((= 0 dd) (A163355 r)) ((= (+ 1 (modulo i 2)) dd) (+ (expt 4 i) (A163355 (A057300 r)))) ((= 3 dd) (+ (* 2 (expt 4 i)) (A163355 (A057300 r)))) (else (+ (* 3 (expt 4 i)) (A163355 (- (expt 4 i) 1 r)))))))
(PARI)
A057300(n) = { my(t=1, s=0); while(n>0, if(1==(n%4), n++, if(2==(n%4), n--)); s += (n%4)*t; n >>= 2; t <<= 2); (s); };
A163355(n) = if(!n, n, my(i = (#binary(n)-1)\2, f = 4^i, d = (n\f)%4, r = (n%f)); if(((1==d)&&!(i%2))||((2==d)&&(i%2)), f+A163355(A057300(r)), if(3==d, f+f+A163355(A057300(r)), (3*f)+A163355(f-1-r)))); \\ Antti Karttunen, Apr 14 2018
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CROSSREFS
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In range [A000302(n-1)..A024036(n)] of this permutation, the number of cycles is given by A163910, number of fixed points seems to be given by A147600(n-1) (fixed points themselves: A163901). Max. cycle sizes is given by A163911 and LCM's of all cycle sizes by A163912.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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