Euler Numbers Modulo Powers of Two and Arnold's Sequence: Proof of Two Conjectures of Ramassamy

Let E_n be the Euler up/down numbers, Σ_n E_n x^n/n! = sec x + tan x, and let e_{n,i} be the Entringer numbers, which form the Seidel–Entringer–Arnold triangle. Arnold observed, without proof, that the least 2-adic valuation m_i of the entries on the i-th diagonal of this triangle is weakly increasing in i, and he introduced the sequence u_k = max{i : m_i < k}. In 2017 Ramassamy conjectured that the sequence (E_n mod 2^k)_{n≥0} is periodic from the index u_k on but not from any earlier index, that its least period is 2^k for k ≠ 2 and 2 for k = 2, and that (u_k) is the f-transform of (2,4,4,4) for an explicit doubling map f. We prove these two conjectures. With h(j) = 2j − 2 − v_2(j), the 2-adic valuation of the tangent number E_{2j−1}, we show that m_i = min_{j≥⌈i/2⌉} h(j) for every i, which proves Arnold's observation, and that u_k = 2 max{j : h(j) < k}. The statements about the period, and the value 2 max{j : h(j) < k} of the preperiod, follow quickly from Stern's classical congruence for the Euler numbers and from the valuation of the tangent numbers. The new ingredients are the determination of m_i and u_k and the proof of the f-transform identity, which rests on the self-similarity h(2^{a−1} + r) = h(r) + 2^a for 1 ≤ r < 2^{a−1}. The proofs are elementary, and computations serve only as consistency checks. This is an unrefereed note.