This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A356717 #18 Oct 02 2022 00:22:02
%S A356717 1,29,59,241,445,1691,3089,11629,21211,79745,145421,546619,996769,
%T A356717 3746621,6831995,25679761,46827229,176011739,320958641,1206402445,
%U A356717 2199883291,8268805409,15078224429,56675235451,103347687745,388457842781,708355589819,2662529664049
%N A356717 a(n) is the integer w such that (c(n)^2, -d(n)^2, w) is a primitive solution to the Diophantine equation 2*x^3 + 2*y^3 + z^3 = 11^3, where c(n) = F(n+2) + (-1)^n * F(n-3), d(n) = F(n+3) + (-1)^n * F(n-2) and F(n) is the n-th Fibonacci number (A000045).
%H A356717 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,7,-7,-1,1).
%F A356717 a(n) = (1331 - 2 * A237132(n)^6 + 2 * A228208(n+1)^6)^(1/3).
%F A356717 a(n) = ((1-(-1)^n)/2) * (-5 + 14 * Sum_{k=1..n-1} Fibonacci(4*k-1) + 6 * Sum_{k=0..n-1} Fibonacci(4*k+1)) + ((1+(-1)^n)/2) * (-5 + 14 * Sum_{k=1..n} Fibonacci(4*k-1) + 6 * Sum_{k=0..n-1} Fibonacci(4*k+1)).
%F A356717 a(n) = ((1-(-1)^n)/2) * (-5 + 14 * A081018(n-1) + 6 * A081016(n-1)) + ((1+(-1)^n)/2) * (-5 + 14 * A081018(n) + 6 * A081016(n-1)).
%F A356717 From _Stefano Spezia_, Aug 25 2022: (Start)
%F A356717 G.f.: x*(1 + 28*x + 23*x^2 - 14*x^3 - 5*x^4)/((1 - x)*(1 - 3*x + x^2)*(1 + 3*x + x^2)).
%F A356717 a(n) = a(n-1) + 7*a(n-2) - 7*a(n-3) - a(n-4) + a(n-5) for n > 5. (End)
%e A356717 For n=3, 2 * ((F(5) - F(0))^2)^3 + 2 * (-(F(6) - F(1))^2)^3 + 59^3 = 2 * 25^3 - 2 * 49^3 + 59^3 = 1331, a(3) = 59.
%t A356717 Table[(1331-2*((Fibonacci[n+2]+(-1)^n*Fibonacci[n-3]))^6+2*(Fibonacci[n+3]+(-1)^n*Fibonacci[n-2])^6)^(1/3), {n,28}]
%Y A356717 Cf. A000045, A081016, A081018, A089270, A228208, A237132.
%Y A356717 Cf. also A337929, A354337, A356716.
%K A356717 nonn,easy
%O A356717 1,2
%A A356717 _XU Pingya_, Aug 24 2022