A317206 An alternative tribonacci representation of n: an encoding of the position of n in the A003144, A003145, A003146 table.

0, 1, 2, 12, 3, 112, 22, 13, 1112, 212, 122, 32, 113, 23, 11112, 2112, 1212, 312, 1122, 222, 132, 1113, 213, 123, 33, 111112, 21112, 12112, 3112, 11212, 2212, 1312, 11122, 2122, 1222, 322, 1132, 232, 11113, 2113, 1213, 313, 1123, 223, 133, 1111112, 211112

Offset: 0

Author

N. J. A. Sloane, Aug 09 2018

Keywords

• nonn
• base

Comments

Let T denote the following 4-rowed table, whose rows are n, A = A003144(n), B = A003145(n), C = A003146(n):

n: 1 .2 .3 .4 .5 .6 .7 .8 .9 ...

A: 1 .3 .5 .7 .8 10 ...

B: 2 .6 .9 13 15 19 ...

C: 4 11 17 24 28 35 ...

Set a(0)=0. For n>0, locate n in rows A, B, C of the table, and indicate how to reach that entry starting from column 1. For example, 17 = C(3) = C(A(2)) = C(A(B(1))), so the path to reach 17 is CAB, which we write (encoding A as 1, B as 2, C as 3) as a(17) = 312.

This is an analog of the Wythoff representation of n described in Lang (1996), A189921, and A317208.

References

• W. Lang, The Wythoff and the Zeckendorf representations of numbers are equivalent, in G. E. Bergum et al. (edts.) Application of Fibonacci numbers vol. 6, Kluwer, Dordrecht, 1996, pp. 319-337. [See A317208 for a link.]

Links

• Lars Blomberg, Table of n, a(n) for n = 0..10000
• Wolfdieter Lang, The Tribonacci and ABC Representations of Numbers are Equivalent, arXiv preprint arXiv:1810.09787 [math.NT], 2018.

Crossrefs

See A278038 for the standard tribonacci representation of n.

See A189921 and A317208 for the analogous Wythoff representation of n.

Cf. A317207.

Extensions

Inserted a(10) and a(18) and beyond from Lars Blomberg, Aug 11 2018