### A Combinatorial Approach to Tanny Sequence

*Anita Das*

#### Abstract

The Tanny sequence T(i) is a sequence defined recursively
as
T(i) = T(i - 1 - T(i - 1)) + T(i - 2 - T(i - 2)),
T(0)
= T(1) = T(2) = 1. In the first part of this paper we give
combinatorial
proofs of all the results regarding T(i), that Tanny
proved
in his paper ``A well-behaved cousin of the Hofstadter sequence'',
Discrete
Mathematics, 105(1992), pp. 227-239, using algebraic means. In most
cases
our proofs turn out to be simpler and shorter. Moreover, they give a
``visual''
appeal to the theory developed by Tanny. We also generalize most of
Tanny's
results. In the second part of the paper we present many new results
regarding
T(i) and prove them combinatorially. Given two integers
n
and k, it is interesting to know if T(n) = k
or not. In this paper we characterize such numbers.

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