Metadata
bibkey: forberg2013a002620
authors: Richard R. Forberg; OEIS Foundation Inc.
year: 2013
title: "Forberg's empirical exponential formula for OEIS A002620"
doi: null
url: https://oeis.org/A002620/internal
claim: "For positive integers n, the floor of n/(exp(4/n)-1) equals the shifted quarter-square a(n-1)."
strata_touched:
- D5/S3/Analytic/SeriesInequalities/ForbergExponentialFloor
license: citation-only
triage: anchor
Verified locator
Source: https://oeis.org/A002620/internal
Machine-readable entry: https://oeis.org/search?q=id:A002620&fmt=json
Revision history: https://oeis.org/history?seq=A002620
Revision 880 has source timestamp 2026-10-21T11:11:28-04:00. The independently
authenticated JSON payload has SHA-256
ac674d2f66044fb747bee82995293f0a3f285677bfc2e220e390e28bf63a4888.
The formula section attributes the following line to Richard R. Forberg:
Empirical: a(n-1) = floor(n/(e^(4/n)-1)). - Richard R. Forberg, Jul 24 2013
Mathematical statement
The entry defines the quarter-squares by
a(m) = floor(m/2)*ceiling(m/2) = floor(m^2/4), with offset zero.
For every natural number n satisfying 1 <= n, the asserted identity is
floor(n/(exp(4/n)-1)) = floor((n-1)^2/4).
All arithmetic inside the floors is real arithmetic, with n coerced before
division and subtraction; exp is the real exponential and both floors are
integer-valued. The endpoint n=1, both parities and unbounded n are included.
The analytic argument places the quotient, for real r >= 2, between
(r-1)^2/4 and r^2/4-r/2+1, with a strict upper inequality. A logarithmic
series bound supplies the lower inequality and the quadratic exponential
bound supplies the upper inequality. Parity identifies the common integer
floor; the endpoint n=1 is handled separately.
Literature boundary
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