A225237 Numbers n such that n occurs within its base 2 representation regarded as a fixed necklace.

0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1100, 1101, 1111, 10000, 10001, 10011, 10100, 10101, 10111, 11000, 11001, 11100, 11101, 100000, 100001, 101000, 101010, 101100, 101101, 101111, 110000, 110001, 110101, 110110, 111011, 111100, 111101, 1000000

Offset: 1

Author

Douglas Latimer, May 04 2013

Keywords

• nonn
• base

Comments

Every power of 10 occurs in this sequence.

Examples

			10 (in base 10) = 1010 (in base 2). Regarding this base 2 representation as a fixed necklace, we can list characters in the order 1010 by starting with the first character. In this listing 10 occurs ({10}10). Thus 10 is in the sequence.
111011 (in base 10) = 11011000110100011 (in base 2). Regarding this base 2 representation as a fixed necklace, we can list characters in the order 11110110001101000 by starting with the characters "11" at the end of the base 2 representation. In this listing 111011 occurs (1{111011}0001101000). Thus 111011 is in the sequence.
		

Links

• Eric W. Weisstein, MathWorld: Necklace

Crossrefs

A038102 is a subsequence of this. For the terms of the sequence not found in A038102, see A225238.

Programs

PARI

{inseq(w)=local(bw,mm,texp,btod,bigb,lbb,swsq,ii);
bw=binary(w);
mm=length(bw); texp=0; btod=0;
forstep(i=mm, 1, -1, btod=btod+bw[i]*10^texp; texp++);
bigb=binary(btod); lbb=length(bigb);
for(k=0, lbb - 1 , swsq=1;
for(j=1, mm, ii=(j+k)%lbb; if(ii==0, ii=lbb);
if(bw[j]!=bigb[ii], swsq=-1));
if(swsq==1, break)
); if(swsq==1,swsq=btod);
return(swsq)}
{ptd=0; for(w=0, 10^9, jj=inseq(w); if(jj>=0, ptd++; print1(jj,", "); if(ptd>39,break)))}