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Exact families and local OEIS comparisons

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The komorebi_math.sequences domain provides exact family generation and comparison with explicit local sequence corpora. These operations work offline and need no additional dependency.

Operation Package implementation
Bernoulli numbers and signed Euler coefficients arithmetic.euler, sequences.catalog
Fibonacci and Lucas recurrences and companion jumps arithmetic.recurrence
Generic Hurwitz products and inversion arithmetic.hurwitz
Ordinary and normalized formal series, reciprocals, powers generating.series, generating.finite_series
Catalan, Bell, partitions, Apéry, and automatic sequences sequences.catalog, generating.sequences
Residue channels, refinement, and polyphase reconstruction refinement.residue, refinement.polyphase

LinearRecurrence orders companion states from oldest to newest and uses x[n+d] = sum(c[i] * x[n+i]). The convenience functions in this namespace use the same convention.

Exact families

from komorebi_math.sequences import (
    reciprocal_coefficients, reciprocal_sequence,
    universal_denominator_coefficients,
    kbonacci_sequence, kbonacci_closed_form,
    fubini_sequence, derangement_sequence,
)

assert reciprocal_coefficients(2, 6) == (1, 1, 5, 61, 1385, 50521)
assert kbonacci_sequence(3).first(7) == (0, 0, 1, 1, 2, 4, 7)
assert kbonacci_closed_form(3, 6) == 7
assert fubini_sequence().first(5) == (1, 1, 3, 13, 75)
assert derangement_sequence().first(6) == (1, 0, 1, 2, 9, 44)

For spacing N and nonnegative integer power r, the normalization is

\[ D_N(t)=\sum_{q\geq0}\frac{(-1)^q t^{Nq}}{(Nq)!},\qquad D_N(t)^{-r}=\sum_{n\geq0}\frac{A_N^{[r]}(n)t^{Nn}}{(Nn)!}. \]

reciprocal_coefficients(N, count, power=r) returns exactly count compressed coefficients starting at n=0. Power zero gives (1,0,0,...). universal_denominator_coefficients(N, degree) returns dense EGF coefficients through the inclusive ordinary degree, including the zero residue lanes. It does not return ordinary Taylor coefficients. The compressed inversion recurrence uses Python integers throughout. The existing dense Hurwitz implementation provides an independent normalization control.

The k-bonacci family has k-1 initial zeros and then one. Its finite binomial formula is tested against the recurrence and modular companion execution. Fubini numbers use ordered set partitions. Derangements start at D_0=1. All integer sizes, offsets, and terms reject booleans and fractional coercions.

Explicit local corpora

from komorebi_math.sequences import (
    OEISRecord, compare_oeis_prefix, find_local_oeis_matches, load_local_oeis,
)

reference = OEISRecord("A000045", (1, 1, 2, 3, 5, 8), offset=1)
match = compare_oeis_prefix((2, 3, 5), reference, query_offset=3, minimum_terms=3)
assert match.passed
assert match.index_alignment_verified

# An explicit file can be internal-format text, stripped text, or archive JSON.
# .gz files are supported. No download occurs.
# corpus = load_local_oeis("dated_snapshot.json", format="json")
# matches = find_local_oeis_matches(query_terms, corpus, query_offset=0)

parse_oeis_internal reads %S/%T/%U fields, preferring signed %V/%W/%X fields when supplied. Continuations are assembled in field order. %N provides the name and %O supplies the starting index. Missing offsets remain None. Lines that do not match the internal field syntax are ignored, including lines with malformed identifiers. Inspect the returned records before comparing them. parse_oeis_stripped reads the terms and optional names. The stripped format contains no index offset, so the loader preserves an unknown offset.

Without query_offset, matching compares the first stored positions and does not claim verified index alignment. With a declared query offset, the matcher aligns actual indices. A reference with an unknown offset cannot validate an explicit indexed query. A query beginning before the available reference is reported as incomplete rather than silently dropping its leading terms.

agrees_on_overlap means every available compared term agrees and the minimum length is met. complete_query_match, and its passed alias, additionally mean that the entire supplied query was compared. If a reference ends early, the status is reference-exhausted-prefix and passed is false. Candidate search can retain that incomplete evidence, but excludes any known mismatch even if an earlier prefix met the minimum. Every result concerns a finite window. Neither a match nor a failed local search establishes infinite identity, originality, priority, or absence from the full OEIS.

Local data and licensing

No OEIS corpus or atlas is bundled with the library. Users load a local snapshot explicitly, and JSON provenance is retained on LocalOEISCorpus.provenance. The original software is licensed under MIT. OEIS data are attributed to the OEIS Foundation under CC BY-SA 4.0, separate from the software license. A software grant does not change the terms of separately supplied data.