First, we prove explicit formula for the symmetric sum (,) + (,) which is a new reciprocity law for the sums above. This formula can be seen as a complement to the Bettin-Conrey result [13, Theorem 1]. Second, we establish an asymptotic formula for (,). Finally, by use of continued fraction theory, we give a formula for (,) in terms of continued fraction of .

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In this paper, we show that there is at most one value of the positive integer X participating in the Pell equation X 2 − dY 2 = k , where k ∈ {±1, ±4} , which is a Padovan number, with a few exceptions that we completely characterize.

International audience In this paper, we find all integers c having at least two representations as a difference between a Pell number and a power of 2.

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