The Alphabet as a Sequence Engine: A Census of What Letter Names Generate, and What the OEIS Has Never Seen

TL;DR

Applying a fixed recursive expansion rule to letter names across 36 writing systems yields 1,056 letter-sequences that collapse into 402 distinct integer sequences, of which 14 match classical OEIS identities term-for-term and 388 appear in no catalogue despite satisfying explicit linear recurrences. Two random-baseline controls (0 of 250 and 0 of 100 simulations) support the claim that natural scripts form unusually small name-graph cores.

Key findings

• Of 1,056 letter-sequences across 36 scripts, 402 are distinct; 14 are OEIS-catalogued (verified term-for-term up to 90 published terms) and 388 are uncatalogued, each satisfying an explicit integer linear recurrence.
• Growth rates cluster on a small set of algebraic numbers: 78 sequences at φ² = 2.618…, 44 at 2+√2 = 3.414…, 27 at 1+√2 = 2.414…, 20 near 3.1416 (English), 19 near 4.21 (Ogham), 19 at exactly 2, 8 at exactly 3 (Korean), and 30 flat (rate 1).
• Recurrence orders run from 1 to more than 14; the most common are 13 (47 sequences), 5 (41), 4 (29), and 11 (23), with 176 sequences needing order above 14.
• Verified identities include powers of two (A000079, from Hebrew bet/tet/mem/nun and the same naming pattern in Phoenician, Aramaic, Syriac, Arabic, Japanese-pronounced Latin, Sanskrit, Amharic), Fibonacci bisection (A001906, Ge'ez and Thai), odd-index Lucas numbers (A002878, two Thai letters), the Pell family (A094723, A048739, Arabic), Mersenne numbers (A000225), and the bare counting numbers (one Syriac letter group).
• Controls: 250 random 26-letter alphabets all entangled into one giant loop (core sizes 21–26, none below 21); 100 random matched-shape alphabets produced no English-style spectrum split (0 of 100).

Methods

• Census design: 36 writing systems totaling 1,056 letters, selected for documented historical names and stylistic diversity; all "failures" retained, including 8 scripts matching nothing in the OEIS (seven formed massive tangled graphs, Georgian collapsed) and several trivial flat sequences.
• Each letter's sequence is generated by a fixed recursive expansion of its name; the script's incidence matrix drives the counts, and the growth rate is the Perron eigenvalue of that matrix. Sequences were computed to 500 terms in exact integer arithmetic.
• Verification: sequences audited against the OEIS, keeping only identities holding term-by-term across the entire available OEIS entry (up to 90 terms) under an exact recurrence check; two random-baseline controls (250 and 100 simulations) tested core size and the English E5 spectrum split.

Limitations

• The paper states no formal limitations section. It explicitly disclaims importance for the uncatalogued sequences: "We make no claim that they are important; we claim only that they are generated, that their recurrences are known, and that no catalogue records them."
• Evident from the text: the random-baseline controls use 26-letter alphabets of matched name length (2–5 letters per name), so their generality to other script sizes is untested; the "name-structure conjecture" is described as supported by the core-size control but is not proven.
• Evident from the text: the census is released as a working paper, and the OEIS comparison is bounded by published record lengths (up to 90 terms) rather than the full 500 generated terms.