bibkey: bala2023a000364 authors: Peter Bala year: 2023 title: “OEIS A000364, Euler (secant) numbers” doi: null url: https://oeis.org/A000364 claim: “Euler (or secant or "Zig") numbers: e.g.f. (even powers only) sec(x) = 1/cos(x). Conjecture: taking the sequence [a(n) : n >= 1] modulo an integer k gives a purely periodic sequence with period dividing phi(k). For example, the sequence taken modulo 21 begins [1, 5, 19, 20, 16, 2, 1, 5, 19, 20, 16, 2, 1, 5, 19, 20, 16, 2, 1, 5, 19, …] with an apparent period of 6 = phi(21)/2. - Peter Bala, May 08 2023” strata_touched:

• D5/S1/Recurrence/Periodic/SecantNumberPurePeriodRefutation license: citation-only triage: anchor

OEIS A000364

Peter Bala’s comment of May 8, 2023 states the pure periodicity conjecture quoted above. The entry defines the Euler secant numbers, with offset 0, by a(0) = 1 and, for positive n, a(n) = Sum_{j=1..n} (-1)^(j+1) * C(2*n,2*j) * a(n-j). This is the coefficient recurrence of sec(x) * cos(x) = 1. The first published values are 1, 1, 5, 61, 1385, 50521, 2702765, 199360981, 19391512145, 2404879675441.

The formal definition uses this recurrence, reindexed by replacing j with j+1, and proves its uniqueness. Reduction modulo 27 commutes with the recurrence. The residues at indices 1 and 19 are respectively 1 and 10. Since phi(27) = 18, these two residues refute pure periodicity with any positive period dividing phi(27).

The neighboring comment cites Knuth and Buckholtz (1967, Theorem 2): the sequence starting at index 1 is periodic modulo every odd prime, with minimal period (p-1)/2 for p == 1 (mod 4) and p-1 for p == 3 (mod 4). The counterexample at the composite modulus 27 does not contradict that statement. No assertion of eventual periodicity is proved here.

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• URL: https://oeis.org/A000364
• Locator: COMMENTS, Peter Bala, May 08 2023; FORMULA and DATA.