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

Numbers n such that A004117(n) is odd. [From Pontus von Broemssen (pontus.von.bromssen(AT)gmail.com), Nov 25 2008]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

For n>=1, a(n)=a(n-1)+(4^t+2)/6, where t is such that 2^t||2n,or t=A007814(2n). a(n)=(145812(n+1)-1)/2 [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Nov 07 2008]

To get a(n), write n as Sum b_j*2^j, then a(n) = Sum b_j*2^(2j). The Diophantine equation a(k)+2a(l)=n has the unique solution: k=Sum b_(2j)*2^j, l=Sum b_(2j+1)*2^j. [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Nov 10 2008]

If a(k)*a(l)=a(m), then k*l=m (inverse, generally speaking, is not true) [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Nov 21 2008]

If n=27, then b_0=1, b_1=1, b_2=0, b_3=1, b_4=1. Therefore a(27)=4^4+4^3+4+1=325; k=b_0+b_2*2+b_4*2^2=5, l=b_1+b_3*2=3, such that a(5)=17, a(3)=5 and 27=17+2*5. [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Nov 10 2008]

(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))", "))

For generating functions Prod_{k>=0} (1+a*x^(b^k)) for the following values of (a,b) see: (1,2) A000012 and A000027, (1,3) A039966 and A005836, (1,4) A151666 and A000695, (1,5) A151667 and A033042, (2,2) A001316, (2,3) A151668, (2,4) A151669, (2,5) A151670, (3,2) A048883, (3,3) A117940, (3,4) A151665, (3,5) A151671, (4,2) A102376, (4,3) A151672, (4,4) A151673, (4,5) A151674.

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.

Sequence in context: A085768 A166304 A078713 this_sequence A081345 A137527 A024854

Adjacent sequences: A000692 A000693 A000694 this_sequence A000696 A000697 A000698

nonn,nice,easy

N. J. A. Sloane (njas(AT)research.att.com).