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 A219014 #17 Jul 20 2025 08:59:47
%S A219014 6,6726,13765255184676885126
%N A219014 Numerators in a product expansion for sqrt(2).
%C A219014 a(3) has 96 digits and a(4) has 479 digits.
%C A219014 Iterating the algebraic identity sqrt(1 + 4/x) = (1 + 2*(x + 2)/(x^2 + 3*x + 1)) * sqrt(1 + 4/(x*(x^2 + 5*x + 5)^2)) produces a rapidly converging product expansion sqrt(1 + 4/x) = Product_{n >= 0} (1 + 2*a(n)/b(n)), where a(n) and b(n) are integer sequences when x is a positive integer.
%C A219014 The present case is when x = 4. The denominators b(n) are in A219015. See also A219010 (x = 1) and A219012 (x = 2).
%H A219014 Amiram Eldar, <a href="/A219014/b219014.txt">Table of n, a(n) for n = 0..4</a>
%F A219014 Let tau = 3 + 2*sqrt(2). Then a(n) = tau^(5^n) + 1/tau^(5^n).
%F A219014 Recurrence equation: a(n+1) = a(n)^5 - 5*a(n)^3 + 5*a(n) with initial condition a(0) = 6.
%e A219014 The first two terms of the product give 18 correct decimal places of sqrt(2): (1 + 2*6/29)*(1 + 2*6726/45232349) = 1.41421 35623 73095 048(5...).
%t A219014 RecurrenceTable[{a[n+1] == a[n]^5 - 5*a[n]^3 + 5*a[n], a[0] == 6}, a, {n, 0, 3}] (* _Amiram Eldar_, Jul 20 2025 *)
%Y A219014 Cf. A156035, A219010, A219012, A219015.
%K A219014 nonn,easy
%O A219014 0,1
%A A219014 _Peter Bala_, Nov 09 2012