A163356 Inverse permutation to A163355, related to Hilbert's curve in N x N grid.

0, 1, 3, 2, 8, 10, 11, 9, 12, 14, 15, 13, 7, 6, 4, 5, 16, 18, 19, 17, 20, 21, 23, 22, 28, 29, 31, 30, 27, 25, 24, 26, 48, 50, 51, 49, 52, 53, 55, 54, 60, 61, 63, 62, 59, 57, 56, 58, 47, 46, 44, 45, 39, 37, 36, 38, 35, 33, 32, 34, 40, 41, 43, 42, 128, 130, 131, 129, 132, 133

Offset

0,3

Links

A. Karttunen, Table of n, a(n) for n = 0..65535

Index entries for sequences that are permutations of the natural numbers

Formula

a(0) = 0, and provided that d=1, 2 or 3, then a((d*(4^i))+r) = (((2+(i mod 2))^d mod 5)-1) * [either A024036(i) - a(r), if d is 3, and A057300(a(r)) in other cases].

From Antti Karttunen, Apr 14 2018: (Start)

A059905(a(n)) = A059253(n).

A059906(a(n)) = A059252(n).

a(n) = A000695(A059253(n)) + 2*A000695(A059252(n)).

(End)

Prog

(MIT Scheme:)
(define (A163356 n) (if (zero? n) n (let* ((i (floor->exact (/ (A000523 n) 2))) (d (modulo (floor->exact (/ n (expt 4 i))) 4)) (r (modulo n (expt 4 i)))) (+ (* (-1+ (modulo (expt (+ 2 (modulo i 2)) d) 5)) (expt 4 i)) (cond ((= 3 d) (- (expt 4 i) 1 (A163356 r))) (else (A057300 (A163356 r))))))))
(Another, iterative version): (define (A163356v2 n) (let loop ((z 0) (n n) (i 0)) (let ((dd (modulo n 4))) (cond ((zero? n) z) ((= 0 dd) (loop z (floor->exact (/ n 4)) (+ i 2))) ((= 2 dd) (loop (+ (* 3 (expt 2 i)) (A057300 z)) (floor->exact (/ n 4)) (+ i 2))) ((= 1 dd) (loop (+ (expt 2 (+ i (floor->exact (/ (modulo i 4) 2)))) (A057300 z)) (floor->exact (/ n 4)) (+ i 2))) (else (loop (+ (expt 2 (+ i (- 1 (floor->exact (/ (modulo i 4) 2))))) (- (expt 2 i) z 1)) (floor->exact (/ n 4)) (+ i 2)))))))
(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); };
A163356(n) = if(!n, n, my(i = (#binary(n)-1)\2, f = 4^i, d = (n\f)%4, r = (n%f)); (((((2+(i%2))^d)%5)-1)*f) + if(3==d, f-1-A163356(r), A057300(A163356(r)))); \\ Antti Karttunen, Apr 14 2018

Crossrefs

Inverse: A163355.

Second and third "powers": A163906, A163916. See also A059252-A059253.

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.

Cf. A059252, A059253, A059905, A059906.

Cf. also A302844, A302846, A302781.

Sequence in context: A371998 A363816 A060921 * A209360 A095013 A094188

Adjacent sequences: A163353 A163354 A163355 * A163357 A163358 A163359

Keyword

nonn

Author

Antti Karttunen, Jul 29 2009

Extensions

Links to further derived sequences and a nicer Scheme function & formula added by Antti Karttunen, Sep 21 2009

Status

approved