0,3

Numbers whose set of base 4 digits is {0,1} - Ray Chandler (rayjchandler(AT)sbcglobal.net), Aug 3 2004

Numbers n such that sum of base 2 digits of n = sum of base 4 digits of n - Clark Kimberling (ck6(AT)evansville.edu).

Numbers having the same representation in both binary and negabinary (A039724) - Eric Weisstein (eric(AT)weisstein.com)

This sequence has many other interesting and useful properties. Every integer n corresponds to a unique pair i,j with n=a(i)+2a(j) (i=A059905(n), j=A059906(n)) - see A126684.. Every list of numbers L=[L1,L2,L3...] can be encoded uniquely by "recursive binary interleaving", where f(L)=a(L1)+2*a(f([L2,L3...])) with f([])=0. Yet another description is "Numbers whose base 4 representation consists of only 0's and 1's". - Marc LeBrun (mlb(AT)well.com), Feb 07 2001

Additional comments from Marc LeBrun (mlb(AT)well.com), Mar 24 2005: This may be described concisely using the "rebase" notation b[n]q, which means "replace b with q in the expansion of n", thus "rebasing" n from base b into base q. The present sequence is 2[n]4. Many interesting operations (e.g. 10[n](1/10) = digit reverse, shifted) are nicely expressible this way. Note that q[n]b is (roughly) inverse to b[n]q. It's also natural to generalize the idea of "basis" so as to cover the likes of F[n]2, the so-called "fibbinary" numbers (A003714), and provide standard ready-made images of entities obeying other arithmetics, say like GF2[n]2 (e.g. primes = A014580, squares = the present sequence, etc).

a(n) is also equal to the product n X n formed using carryless binary multiplication (A059729, A063010). - Henry Bottomley (se16(AT)btinternet.com), Jul 03 2001

J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.

N. G. de Bruijn, Some direct decompositions of the set of integers, Math. Comp., 18 (1964), 537-546.

L. Moser, An application of generating series, Math. Mag., 35 (1962), 37-38.

T. D. Noe, Table of n, a(n) for n=0..1023

Joerg Arndt, Fxtbook

J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.

Eigen, S. J.; Ito, Y.; and Prasad, V. S., Universally bad integers and the 2-adics, J. Number Theory 107 (2004), 322-334.

R. Stephan, Some divide-and-conquer sequences ...

R. Stephan, Table of generating functions

R. Stephan, Divide-and-conquer generating functions. I. Elementary sequences

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

G.f. 1/(1-x) * Sum(k>=0, 4^k*x^2^k/(1+x^2^k)). - Ralf Stephan, Apr 27 2003

n such that the coefficient of x^n is > 0 in prod (k>=0, 1+x^(4^k)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Jul 29 2003

(PARI) d(n, k)=if(n<=0, [ ], concat(d(n\k, k), n%k)); /* vector of digits of n in base k */ sd(n, k)=Set(d(n, k)) /* set of digits of n in base k */

(PARI) for(n=0, 100, l=if(!n, 0, floor(log(n)/log(2))):print1(sum(k=0, l, binary(n)[k+1]*4^(l-k))", "))

Diagonal of A048720, second column of A048723.

Cf. A059884, A059901, A059904, A059905, A059906, A005836, A007088, A033042-A033052.

Cf. A126684.

A062880[n] = 2*a[n]; A001196[n] = 3*a[n].

Row 4 of array A104257.

Adjacent sequences: A000692 A000693 A000694 this_sequence A000696 A000697 A000698

Sequence in context: A119562 A085768 A078713 this_sequence A081345 A137527 A024854

nonn,nice,easy

njas