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A384106 Numbers representable as the sum of 2 cubes in at least 2 ways generated by a parameterized formula involving (7+4*sqrt(3))^n and (7-4*sqrt(3))^n.

%I A384106 #34 Jun 30 2025 18:18:35
%S A384106 1009736,2714690888,7334904115448,19818905563705976,
%T A384106 53550675461437475048,144693905277386048024168,
%U A384106 390962878508814502873889816,1056203940519850679825934312168,2853755704387709706549646191448888,7710144396612746633517746345789261976
%N A384106 Numbers representable as the sum of 2 cubes in at least 2 ways generated by a parameterized formula involving (7+4*sqrt(3))^n and (7-4*sqrt(3))^n.
%C A384106 A rapidly growing sequence of integers, each equal to x(n)^3 + y(n)^3 = u(n)^3 + w(n)^3 for distinct positive integers x(n), y(n), u(n), w(n), generated from a parameterized expression. Values omit small classical examples (like 1729) and begin at much larger values and is therefore a parameterized subset of solutions to A001235.
%H A384106 Jamal Agbanwa, <a href="https://doi.org/10.6084/m9.figshare.29083724">A Closed-Form Symbolic Generator: A^n + B^n = C^n + D^n, n = 2, 3</a>, Preprint, 2025. See also <a href="https://arxiv.org/abs/2506.19173">arXiv:2506.19173</a> [math.GM], 2025, p. 8.
%F A384106 a(n) = x(n)^3 + y(n)^3 = u(n)^3 + w(n)^3 where:
%F A384106   x(n) = (-6 + (15 - 7*sqrt(3))*(7 - 4*sqrt(3))^n + (15 + 7*sqrt(3))*(7 + 4*sqrt(3))^n)/4 + 3,
%F A384106   y(n) = (-18 + (7 - 5*sqrt(3))*(7 - 4*sqrt(3))^n + (7 + 5*sqrt(3))*(7 + 4*sqrt(3))^n)/4,
%F A384106   u(n) = (-6 + (15 - 7*sqrt(3))*(7 - 4*sqrt(3))^n + (15 + 7*sqrt(3))*(7 + 4*sqrt(3))^n)/4, abd
%F A384106   w(n) = (-18 + (7 - 5*sqrt(3))*(7 - 4*sqrt(3))^n + (7 + 5*sqrt(3))*(7 + 4*sqrt(3))^n)/4 + 9.
%e A384106 For n = 7, a(7) = x(n)^3 + y(n)^3 = ((-6 + (15 - 7*sqrt(3))*(7 - 4*sqrt(3))^7 + (15 + 7*sqrt(3))*(7 + 4*sqrt(3))^7)/4 + 3)^3 + ((-18 + (7 - 5*sqrt(3))*(7 - 4*sqrt(3))^7 + (7 + 5*sqrt(3))*(7 + 4*sqrt(3))^7)/4)^3 = 390962878508814502873889816.
%Y A384106 Cf. A011541, A018850.
%Y A384106 Subset of A001235.
%K A384106 nonn
%O A384106 1,1
%A A384106 _Jamal Agbanwa_, May 19 2025