A006288 Loxton-van der Poorten sequence: base-4 representation contains only -1, 0, +1.

0, 1, 3, 4, 5, 11, 12, 13, 15, 16, 17, 19, 20, 21, 43, 44, 45, 47, 48, 49, 51, 52, 53, 59, 60, 61, 63, 64, 65, 67, 68, 69, 75, 76, 77, 79, 80, 81, 83, 84, 85, 171, 172, 173, 175, 176, 177, 179, 180, 181, 187, 188, 189, 191, 192, 193, 195, 196, 197, 203, 204, 205, 207, 208, 209

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Examples

			1*4^2 + 0*4^1 + (-1)*4^0 = 15, so 15 is in sequence.
		

References

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

Links

• J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197. [DOI]
• J. H. Loxton and A. J. van der Poorten, An Awful Problem About Integers in Base Four, Acta Arithmetica, volume 49, 1987, pages 193-203.
• J. H. Loxton and A. J. van der Poorten, Arithmetic properties of automata: regular sequences, J. Reine Angew. Math. 392 (1988), 57-69. Also second author's copy. See section 1 example.
• A. J. van der Poorten, An Awful Problem about Integers in Base Four (and abstract), slides of a talk at University of Sydney CeNTRe for Number Theory Research, 2007.

Crossrefs

Cf. A344892 (in base 4), A344893 (morphism).

Cf. A023713, A023717.

Programs

PARI

a(n)=if(n<2,n>0,4*a((n+1)\3)+(n+1)%3-1) \\ Ralf Stephan, Jan 19 2014

PARI

a(n) = my(v=digits(n,3),prev=0); forstep(i=#v,1,-1, prev=(v[i]+=(v[i]>(prev<2)))); fromdigits(v,4); \\ Kevin Ryde, Jun 03 2021

Formula

Recurrence: a(3n) = 4a(n), a(3n-1) = 4a(n)-1, a(3n+1) = 4a(n)+1, starting 0,1. - Ralf Stephan, Jan 19 2014

Extensions

Offset changed to 0 and example added by Ralf Stephan, Jan 19 2014