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%I #23 Apr 08 2021 20:06:07
%S 0,1,2,5,4,3,6,7,8,15,16,17,14,13,12,9,10,11,18,19,20,23,22,21,24,25,
%T 26,47,46,45,48,49,50,53,52,51,44,43,42,39,40,41,38,37,36,29,28,27,30,
%U 31,32,35,34,33,54,55,56,59,58,57,60,61,62,69,70,71,68,67,66,63,64,65
%N Self-inverse permutation of the integers for constructing the Peano curve in an N X N grid.
%C The integers [0,(3^k)-1] are confined to range [0,(3^k)-1].
%C From _Kevin Ryde_, Sep 04 2020: (Start)
%C To calculate a(n), write n in ternary digits n[k],..,n[0], where n[0] is the least significant digit. Then the ternary digits of a(n) are a[j] = k^{n[j+1]+n[j+3]+...}(n[j]) where Peano's complement operator k^{s}(d) = d if s even or 2-d if s odd.
%C A single complement is k(d) = 2-d and the "exponent" is repeats k(k(k(...))). Sum s = n[j+1] + n[j+3] + ... is every second digit above j, so digit j flips 0 <-> 2 when an odd number of odd digits (1's) among these. The complement does not change digit parity so a second transformation re-complements back to the original digits and so self-inverse a(a(n)) = n.
%C Peano's curve is formed by digits of a(n) put alternately to x and y coordinates, so a(n) maps between the Peano curve the ternary Z-order curve per the formulas in A163528, A163529.
%C (End)
%H Antti Karttunen, <a href="/A163332/b163332.txt">Table of n, a(n) for n = 0..59048</a>
%H Giuseppe Peano, <a href="https://doi.org/10.1007/BF01199438">Sur une courbe, qui remplit toute une aire plane</a>, Mathematische Annalen, volume 36, number 1, 1890, pages 157-160. Also <a href="https://eudml.org/doc/157489">EUDML</a> (link to GDZ).
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F a(n) = A163327(A163333(A163327(n))).
%t a[n_] := FromDigits[With[{d = Reverse@IntegerDigits[n, 3]}, Reverse@Table[
%t If[EvenQ@Total@d[[j+1 ;; ;; 2]], d[[j]], 2-d[[j]]], {j, Length@d}]], 3];
%t Array[a, 100] (* _Andrey Zabolotskiy_, Apr 08 2021, after _Kevin Ryde_ *)
%o (MIT Scheme:)
%o (define (A163332 n) (let loop ((z 0) (n n) (i 0)) (let ((x (modulo n 3)) (y (modulo (floor->exact (/ n 3)) 3))) (cond ((zero? n) z) ((and (= 1 x) (= 1 y)) (loop (+ (* 4 (expt 3 i)) (complement-i-lsts z i)) (floor->exact (/ n 9)) (+ i 2))) ((= 1 x) (loop (+ (* (+ (* y 3) 1) (expt 3 i)) (complement-i-oddpos-lsts z (/ i 2))) (floor->exact (/ n 9)) (+ i 2))) ((= 1 y) (loop (+ (* (+ 3 (- 2 x)) (expt 3 i)) (complement-i-evenpos-lsts z (/ i 2))) (floor->exact (/ n 9)) (+ i 2))) (else (loop (+ (* (+ (* y 3) x) (expt 3 i)) z) (floor->exact (/ n 9)) (+ i 2)))))))
%o (define (complement-i-lsts n i) (if (zero? i) n (+ (- 2 (modulo n 3)) (* 3 (complement-i-lsts (floor->exact (/ n 3)) (-1+ i))))))
%o (define (complement-i-evenpos-lsts n i) (if (zero? i) n (+ (- 2 (modulo n 3)) (* 3 (complement-i-oddpos-lsts (floor->exact (/ n 3)) (-1+ i))))))
%o (define (complement-i-oddpos-lsts n i) (+ (* 3 (complement-i-evenpos-lsts (floor->exact (/ n 3)) i)) (modulo n 3)))
%o (PARI) a(n) = my(v=digits(n,3)); for(start=2,3, my(s=0); forstep(i=start,#v,2, s+=v[i-1]; if(s%2,v[i]=2-v[i]))); fromdigits(v,3); \\ _Kevin Ryde_, Sep 04 2020
%Y Coordinates using this transform: A163528, A163529.
%Y A163334 & A163336 give two variants of the Peano curve in an N X N grid.
%Y Cf. A163355 (Hilbert curve).
%K nonn
%O 0,3
%A _Antti Karttunen_, Jul 29 2009
%E Name corrected by _Kevin Ryde_, Aug 27 2020
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