A006288 Loxton-van der Poorten sequence: base-4 representation contains only -1, 0, +1. (Formerly M2312)

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

0,3

References

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

Links

Table of n, a(n) for n=0..64.

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.

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

Example

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

Prog

(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

Crossrefs

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

Cf. A023713, A023717.

Sequence in context: A224503 A349005 A128920 * A047598 A283773 A214256

Adjacent sequences: A006285 A006286 A006287 * A006289 A006290 A006291

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

Status

approved