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 A219012 #17 Jul 20 2025 08:59:34
%S A219012 4,724,198924689265124,
%T A219012 311489521560905211009923410118036749078665998068304623331823899816643124
%N A219012 Numerators in a product expansion for sqrt(3).
%C A219012 a(4) has 358 digits.
%C A219012 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 the rapidly converging product expansion sqrt(1 + 4/x) = Product_{n >= 0} (1 + 2*a(n)/b(n)); a(n) and b(n) are integer sequences when x is a positive integer.
%C A219012 The present case is when x = 2. The denominators b(n) are in A219013. See also A219010 (x = 1) and A219014 (x = 4).
%H A219012 Amiram Eldar, <a href="/A219012/b219012.txt">Table of n, a(n) for n = 0..4</a>
%F A219012 Let tau = 2 + sqrt(3). Then a(n) = tau^(5^n) + 1/tau^(5^n).
%F A219012 Recurrence equation: a(n+1) = a(n)^5 - 5*a(n)^3 + 5*a(n) with initial condition a(0) = 4.
%F A219012 a(n) = 2*T(5^n,2), where T(n,x) denotes the n-th Chebyshev polynomial of the first kind. - _Peter Bala_, Mar 29 2022
%F A219012 Let b(n) = a(n) - 4. The sequence {b(n)} appears to be a strong divisibility sequence, that is, gcd(b(n),b(m)) = b(gcd(n,m)) for n, m >= 1. - _Peter Bala_, Dec 06 2022
%e A219012 The first three terms of the product give 70 correct decimal places of sqrt(3): (1 + 2*4/11)*(1 + 2*724/523451)*(1 + 2*198924689265124/39571031999225940638473470251) = 1.73205 08075 68877 29352 74463 41505 87236 69428 05253 81038 06280 55806 97945 19330 (0...).
%t A219012 RecurrenceTable[{a[n+1] == a[n]^5 - 5*a[n]^3 + 5*a[n], a[0] == 4}, a, {n, 0, 3}] (* _Amiram Eldar_, Jul 20 2025 *)
%Y A219012 Cf. A019973, A219010, A219014, A219013.
%K A219012 nonn,easy
%O A219012 0,1
%A A219012 _Peter Bala_, Nov 09 2012