

A163355


Permutation of integers for constructing Hilbert curve in N x N grid.


24



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

0,3


LINKS

A. Karttunen, Table of n, a(n) for n = 0..262143
Index entries for sequences that are permutations of the natural numbers


FORMULA

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
= 2*(4^i) + a(A057300(r)), if d=3,
= 3*(4^i) + a((4^i)1r) in other cases.


PROG

(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(f1r)))); \\ Antti Karttunen, Apr 14 2018


CROSSREFS

Inverse: A163356. A163357 & A163359 give two variants of Hilbert curve in N x N grid. Cf. also A163332.
Second and third "powers": A163905, A163915.
In range [A000302(n1)..A024036(n)] of this permutation, the number of cycles is given by A163910, number of fixed points seems to be given by A147600(n1) (fixed points themselves: A163901). Max. cycle sizes is given by A163911 and LCM's of all cycle sizes by A163912.
See also: A163890, A163894, A163902A163903, A163914, A163485, A302843, A302845.
Sequence in context: A324012 A231183 A324661 * A214885 A145747 A055234
Adjacent sequences: A163352 A163353 A163354 * A163356 A163357 A163358


KEYWORD

nonn


AUTHOR

Antti Karttunen, Jul 29 2009


EXTENSIONS

Links to further derived sequences added by Antti Karttunen, Sep 21 2009


STATUS

approved



