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Metadata 
bibkey: oeis2026a400645
authors: Bence Bernáth
year: 2026
title: "OEIS A400645, signed four-lag rounded norm ratio"
doi: null
url: https://oeis.org/A400645
claim: "The consecutive ratio of the signed four-lag rounded norm sequence tends to sqrt(A289917), the largest real root of x^8-x^6-x^4-x^2+1."
strata_touched:
  - D5/S1/Recurrence/OEISA400645
license: citation-only
triage: anchor

Verified locator

Source: https://oeis.org/A400645

Source statement

OEIS A400645, revision 21, NAME field (unpaginated):

a(n+1) = round(sqrt(a(n)^2 + a(n-1)^2 + a(n-2)^2 - a(n-3)^2)).

COMMENTS field, complete paragraph:

Numerical evidence suggests that the ratio of consecutive terms (a(n+1)/a(n)) tends to 1.3122819079… (sqrt(A289917)) which is the largest real root of the asymptotic equation of this sequence: x^8-x^6-x^4-x^2+1. This is related to A116732 where the asymptotic equation is y^4-y^3-y^2-y+1 via y=x^2.

The Mathematica field initializes a[1] == a[2] == a[3] == 0, a[4] == 1. The OFFSET is 1,7. Index zero is unused. Signed subtraction is taken in integers, and an integer radicand has no half-integer square root.

The cross-reference OEIS A289917, revision 48, FORMULA field states:

Equals (1 + sqrt(13) + sqrt(2*sqrt(13) - 2))/4. - Vaclav Kotesovec, Aug 27 2021

Largest real root of x^4 - x^3 - x^2 - x + 1. - Linas Vepstas, Feb 06 2024

Scope

The target is an OEIS assertion; no arXiv source is identified. A400644 uses three positive squares and different initial data. Its sextic limit does not establish this signed four-lag octic limit.

The bounded literature screen is recorded in issue https://github.com/the-omega-institute/trureturing/issues/22857. Its completeness is ASSUMED-UNVERIFIED. The full text of the related Sirvent article, DOI 10.1051/ita/2026034, was inaccessible in that screen; metadata and abstract do not establish or exclude a settlement.

Experiment

The independent checker is in the-omega-institute/trureturing-experiments, commit 9cfc9a4c03fadd43487edc9fb74b7b753fea6d3f, at docs/reports/oeis-a400645-rounded-norm-ratio/check.py. Its SHA-256 is 20c3145f200623eb7391931d874188e0849f111bfec84370212d01bcd3a00a10. The recorded command python3 docs/reports/oeis-a400645-rounded-norm-ratio/check.py exited 0 with ALL_CHECKS_OK; its finite checks support the seed and algebraic calculations, not the infinite convergence assertion.