A219014 -id:A219014 - cp's OEIS frontend

A219010 Numerators in a product expansion for sqrt(5).

3, 123, 28143753123, 17656721319717734662791328845675730903632844218828123

Offset: 0

Peter Bala, Nov 09 2012

a(4) has 262 digits.
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)), where a(n) and b(n) are integer sequences when x is an integer: a(n) satisfies the recurrence a(n+1) = a(n)^5 - 5*a(n)^3 + 5*a(n) with the initial condition a(0) = x + 2 and b(n) satisfies the recurrence b(n+1) = 5/2*(b(n)^4 - b(n)^2)*sqrt(4*b(n) + 5) + b(n)^5 + 15/2*b(n)^4 - 25/2*b(n)^2 + 5 with the initial condition b(0) = x^2 + 3*x + 1.
The present case is when x = 1. The denominators b(n) are in A219011. See also A219012 (x = 2) and A219014 (x = 4).
For another product expansion for sqrt(5) see A002814.

			The first three terms of the product give 51 correct decimal places of sqrt(5): (1 + 2*3/5)*(1 + 2*123/15005)*(1 + 2*28143753123/792070839820228500005) = 2.23606 79774 99789 69640 91736 68731 27623 54406 18359 61152 5(4...).

Amiram Eldar, Table of n, a(n) for n = 0..4

Cf. A000032, A000045, A002814, A104457, A219011, A219012, A219014.

Table[(1 + 5*(Fibonacci[5^n] Fibonacci[5^n + 1] + Fibonacci[5^n - 1]^2))/2, {n, 0, 3}] (* or *)
Table[LucasL[2*5^n], {n, 0, 3}] (* Michael De Vlieger, Jul 30 2017 *)

A219010(n):=1/2*(1 + 5*(fib(5^n)*fib(5^n+1)+(fib(5^n - 1))^2))$
makelist(A219010(n),n,0,3); /* Martin Ettl, Nov 12 2012 */

Let tau = (3 + sqrt(5))/2. Then a(n) = tau^(5^n) + 1/tau^(5^n).
a(n) = (1/2)*(1 + 5*(Fibonacci(5^n)*Fibonacci(5^n + 1) + (Fibonacci(5^n - 1))^2)).
Recurrence equation: a(n+1) = a(n)^5 - 5*a(n)^3 + 5*a(n) with initial condition a(0) = 3.
a(n) = Lucas(2*5^n). - Ehren Metcalfe, Jul 29 2017

A219012 Numerators in a product expansion for sqrt(3).

Original entry on oeis.org

4, 724, 198924689265124, 311489521560905211009923410118036749078665998068304623331823899816643124

Offset: 0

Peter Bala, Nov 09 2012

a(4) has 358 digits.
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.
The present case is when x = 2. The denominators b(n) are in A219013. See also A219010 (x = 1) and A219014 (x = 4).

			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...).

Amiram Eldar, Table of n, a(n) for n = 0..4

Cf. A019973, A219010, A219014, A219013.

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 *)

Let tau = 2 + sqrt(3). Then a(n) = tau^(5^n) + 1/tau^(5^n).
Recurrence equation: a(n+1) = a(n)^5 - 5*a(n)^3 + 5*a(n) with initial condition a(0) = 4.
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
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

A219015 Denominators in a product expansion for sqrt(2).

Original entry on oeis.org

29, 45232349, 189482250299273866821980904657123150749

Offset: 0

Peter Bala, Nov 09 2012

a(3) has 192 digits and a(4) has 957 digits.

The product expansion in question is sqrt(2) = Product_{n = 0..infinity} (1 + 2*A219014(n)/A219015(n)) = (1 + 2*6/29)*(1 + 2*6726/45232349)*....

Alois P. Heinz, Table of n, a(n) for n = 0..4

Cf. A000129, A219011, A219013, A219014.

Table[Fibonacci[5^(n+1),2]/Fibonacci[5^n,2], {n,0,5}] (* G. C. Greubel, Feb 02 2018 *)

a(n) = Pell(5^(n+1))/Pell(5^n), where Pell(n) = A000129(n).

Recurrence equation: a(n+1) = 5/2*(a(n)^4 - a(n)^2)*sqrt(4*a(n) + 5) + a(n)^5 + 15/2*a(n)^4 - 25/2*a(n)^2 + 5 with initial condition a(0) = 29.

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A219010 Numerators in a product expansion for sqrt(5).

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A219012 Numerators in a product expansion for sqrt(3).

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A219015 Denominators in a product expansion for sqrt(2).

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