A163355 - OEIS

%I #12 Nov 22 2023 04:55:50

%S 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,

%T 31,28,24,25,27,26,58,57,59,56,54,53,55,52,60,61,63,62,50,51,49,48,32,

%U 35,33,34,36,37,39,38,46,45,47,44,40,41,43,42,234,235,233,232,236,239

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

%H A. Karttunen, <a href="/A163355/b163355.txt">Table of n, a(n) for n = 0..262143</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%F a(0) = 0,

%F and given d=1, 2 or 3, then a((d*(4^i))+r)

%F = (4^i) + a(A057300(r)), if d=1 and i is even, or if d=2 and i is odd

%F = 2*(4^i) + a(A057300(r)), if d=3,

%F = 3*(4^i) + a((4^i)-1-r) in other cases.

%p A057300 := proc(n)

%p     option remember;

%p     `if`(n=0, 0, procname(iquo(n, 4, 'r'))*4+[0, 2, 1, 3][r+1])

%p end proc:

%p A163355 := proc(n)

%p     option remember ;

%p     local d,base4,i,r ;

%p     if n <= 1 then

%p         return n ;

%p     end if;

%p     base4 := convert(n,base,4) ;

%p     d := op(-1,base4) ;

%p     i := nops(base4)-1 ;

%p     r := n-d*4^i ;

%p     if ( d=1 and type(i,even) ) or ( d=2 and type(i,odd)) then

%p         4^i+procname(A057300(r)) ;

%p     elif d= 3 then

%p         2*4^i+procname(A057300(r)) ;

%p     else

%p         3*4^i+procname(4^i-1-r) ;

%p     end if;

%p end proc:

%p seq(A163355(n),n=0..100) ; # _R. J. Mathar_, Nov 22 2023

%o (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)))))))

%o (PARI)

%o 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); };

%o 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

%Y Inverse: A163356. A163357 & A163359 give two variants of Hilbert curve in N x N grid. Cf. also A163332.

%Y Second and third "powers": A163905, A163915.

%Y 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.

%Y See also: A163890, A163894, A163902-A163903, A163914, A163485, A302843, A302845.

%K nonn

%O 0,3

%A _Antti Karttunen_, Jul 29 2009

%E Links to further derived sequences added by _Antti Karttunen_, Sep 21 2009