A163355 Permutation of integers for constructing Hilbert curve in N x N grid.
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
Offset: 0
Keywords
Links
Crossrefs
Inverse: A163356. A163357 & A163359 give two variants of Hilbert curve in N x N grid. Cf. also A163332.
Programs
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Maple
A057300 := proc(n) option remember; `if`(n=0, 0, procname(iquo(n, 4, 'r'))*4+[0, 2, 1, 3][r+1]) end proc: A163355 := proc(n) option remember ; local d,base4,i,r ; if n <= 1 then return n ; end if; base4 := convert(n,base,4) ; d := op(-1,base4) ; i := nops(base4)-1 ; r := n-d*4^i ; if ( d=1 and type(i,even) ) or ( d=2 and type(i,odd)) then 4^i+procname(A057300(r)) ; elif d= 3 then 2*4^i+procname(A057300(r)) ; else 3*4^i+procname(4^i-1-r) ; end if; end proc: seq(A163355(n),n=0..100) ; # R. J. Mathar, Nov 22 2023
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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(f-1-r)))); \\ Antti Karttunen, Apr 14 2018
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)-1-r) in other cases.
From Alan Michael Gómez Calderón, May 06 2025: (Start)
a((2 + 16^n)*2^(-1 + 4*m)) = 4^(2*(n + m) - 1) + (11*16^m - 2)/3. (End)
Extensions
Links to further derived sequences added by Antti Karttunen, Sep 21 2009