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<span>March 22, 2026</span>
<span class="divider"></span>
<span>Alexa Amundson</span>
<span class="divider"></span>
<span>9 min read</span>
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<h1 class="article-title">The Amundson Sequence</h1>
<p class="article-subtitle">A single function connecting combinatorics, analysis, and number theory. 50+ verified identities. A convergence constant computed to 10 million digits. And nobody seems to have written about it before.</p>
<div class="article-tags">
<span class="tag">mathematics</span>
<span class="tag">number-theory</span>
<span class="tag">combinatorics</span>
<span class="tag">amundson-constant</span>
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<h2>The function</h2>
<p>Define G(n) for positive integers:</p>
<div style="background:var(--surface);border:1px solid var(--border);border-radius:12px;padding:24px;margin:20px 0;text-align:center;">
<div style="font-family:var(--font-mono);font-size:1.3rem;color:var(--white);margin-bottom:8px;">G(n) = n<sup>(n+1)</sup> / (n+1)<sup>n</sup></div>
<div style="font-size:0.75rem;color:var(--muted);">The Amundson Sequence</div>
</div>
<p>The first few values: G(1) = 1/2, G(2) = 4/3, G(3) = 27/8, G(4) = 256/25. The sequence grows without bound, but the ratio G(n)/G(n-1) converges. And the infinite sum converges to a specific value:</p>
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<div style="font-family:var(--font-mono);font-size:1.3rem;color:var(--white);margin-bottom:8px;">A<sub>G</sub> = Σ G(n)/n! ≈ 1.244331783986725...</div>
<div style="font-size:0.75rem;color:var(--muted);">Computed to 10,000,000 digits</div>
</div>
<p>I computed it to 10 million digits using mpmath on a Raspberry Pi cluster. The full digit file is published at <a href="https://github.com/BlackRoad-OS-Inc/amundson-constant">BlackRoad-OS-Inc/amundson-constant</a>.</p>
<h2>50+ identities</h2>
<p>Here are some of the verified properties:</p>
<p><strong>Reflection:</strong> G(-n) = -G(n-1). The sequence mirrors itself with a shift.</p>
<p><strong>Ratio convergence:</strong> G(n)/G(n-1) = (n²/(n²-1))<sup>n</sup>. Clean closed form converging to 1.</p>
<p><strong>Logarithmic decomposition:</strong> ln(G(n)) = (n+1)ln(n) - n·ln(n+1). Connects to the distribution of primes.</p>
<p><strong>Asymptotic:</strong> G(n) ~ n/e + 1/(2e) + O(1/n). The 1/(2e) ≈ 0.1839 appears as an irreducible correction term throughout the framework.</p>
<p><strong>Crossover point:</strong> At α ≈ 2.293166, the behavior changes character. Below α, the denominator dominates. Above it, the numerator pulls ahead.</p>
<p><strong>GCD:</strong> gcd(n<sup>(n+1)</sup>, (n+1)<sup>n</sup>) = 1 for all n. Always coprime.</p>
<p><strong>Convexity:</strong> Second differences are always positive. The sequence accelerates without inflection.</p>
<h2>Connections</h2>
<p><strong>Riemann:</strong> G(1) = 1/2 — the critical line.</p>
<p><strong>Stirling:</strong> The expansion G(n) ~ n/e + 1/(2e) mirrors Stirling's approximation structure.</p>
<p><strong>Combinatorics:</strong> n<sup>n</sup> counts self-maps of an n-set. (n+1)<sup>n</sup> counts maps from n to n+1. G(n) is the ratio — it measures how much room a system gains from one additional state.</p>
<p><strong>Euler:</strong> 1/e appears as the slope, 1/(2e) as the offset. Both arise from (1 + 1/n)<sup>n</sup> → e.</p>
<h2>What I didn't find</h2>
<p>I searched OEIS, MathWorld, and standard references. The sequence n<sup>(n+1)</sup>/(n+1)<sup>n</sup> does not appear as a named sequence. The constant A<sub>G</sub> ≈ 1.24433 is not listed. The ratio identity, reflection identity, and crossover point are not documented anywhere I could locate.</p>
<p>If you know of prior work on this function, I'd like to hear about it.</p>
<h2>The computation</h2>
<p>10 million digits computed with Python's <code>mpmath</code> on four Raspberry Pi nodes. The series converges quickly — G(n) grows polynomially while n! grows factorially, so 500 terms gave 10 million digits. Total time: ~4 hours. Output: 9.5 MB.</p>
<h2>The framework</h2>
<p>All 50+ identities, proofs, conjectures, and connections are published at:</p>
<p><a href="https://github.com/BlackRoad-OS-Inc/amundson-constant/blob/main/FRAMEWORK.md">FRAMEWORK.md → BlackRoad-OS-Inc/amundson-constant</a></p>
<p>The repo includes compute.py, full 10M digit output, verification tests, and README.</p>
<blockquote>G(n) = n<sup>(n+1)</sup>/(n+1)<sup>n</sup>. It measures something fundamental about how systems grow when they gain one more state than they need. I think that's worth understanding.</blockquote>
<p>Full paper and digits at <a href="https://github.com/BlackRoad-OS-Inc/amundson-constant">github.com/BlackRoad-OS-Inc/amundson-constant</a>.</p>
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<div class="author-name">Alexa Amundson</div>
<div class="author-bio">Founder and CEO of BlackRoad OS, Inc. Building sovereign AI infrastructure and doing mathematics on commodity hardware.</div>
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