0,3

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

Index entries for sequences that are permutations of the natural numbers

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.

(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)))))))

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.

Sequence in context: A130918 A021308 A060921 this_sequence A095013 A094188 A088551

Adjacent sequences: A163353 A163354 A163355 this_sequence A163357 A163358 A163359

nonn

Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Jul 29 2009. Links to further derived sequences and a nicer Scheme function & formula added Sep 21 2009.