%I
%S 0,1,2,3,4,5,6,7,3,4,5,6,2,3,4,5,1,2,3,4,5,6,7,8,4,5,6,7,3,4,5,6,2,3,
%T 4,5,6,7,8,9,5,6,7,8,4,5,6,7,3,4,5,6,7,8,9,10,6,7,8,9,5,6,7,8,4,5,6,7,
%U 8,9,10,11,7,8,9,10,6,7,8,9,5,6,7,8,9,10,11,12,8,9,10,11,7,8,9,10,6,7,8,9
%N Number of one bits in binary representation of base i1 expansion of n (where i = sqrt(1)).
%C First differences are usually +1, occasionally 4 (because in base i1 [3]+[7]=(+i)+(i)=0) hence often a(i+j)=a(i)+a(j). Differences terms given here are period16, but for full sequence is actually period256 at least.
%C a(n) is the sum of the digits of n when written in base 4 using digits 0 to 3 (A007608). This is since in Penney's digit substitution for A066321, the base 4 digits 0 to 3 become bit strings of exactly 0 to 3 many 1bits each respectively.  _Kevin Ryde_, Sep 09 2019
%D D. E. Knuth, The Art of Computer Programming. AddisonWesley, Reading, MA, 1969, Vol. 2, p. 172. (Also exercise 16, p. 177, answer, p. 494.)
%H David A. Corneth, <a href="/A066323/b066323.txt">Table of n, a(n) for n = 0..9999</a>
%H Walter Penney, <a href="https://www.nsa.gov/Portals/70/documents/newsfeatures/declassifieddocuments/techjournals/abinarysystem.pdf">A Binary System for Complex Numbers</a>, NSA Technical Journal, 1965.
%H Walter Penney, <a href="https://doi.org/10.1145/321264.321274">A Binary System for Complex Numbers</a>, Journal of the Association for Computing Machinery (JACM), volume 12, number 2, April 1965, pages 247248.
%F a(n) = A000120(A066321(n)) = A007953(A007608(n)).  _Kevin Ryde_, Sep 09 2019
%e A066321(4) = 464 = 111010000 (binary) so a(4) = 4. Or A007608(4) == 130 in base 4 and sum of digits is a(4) = 1+3+0 = 4.
%t a[n_] := Plus @@ Mod[NestWhileList[(#  Mod[#, 4])/4 &, n, # != 0 &], 4]; Array[a, 100, 0] (* _Amiram Eldar_, Mar 22 2021 *)
%o (PARI) a(n) = my(ret=0); while(n, ret+=n%4; n\=4); ret; \\ _Kevin Ryde_, Sep 09 2019
%Y Cf. A066321, A000120, A007608, A342728.
%K nonn,easy,base
%O 0,3
%A _Marc LeBrun_, Dec 14 2001
