A002814 -id:A002814 - 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

A006273 Numerators of a continued fraction for (3+sqrt(13))/2.

Original entry on oeis.org

3, 10, 1297, 2186871697, 10458512317535240383929505297

Offset: 0

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jeffrey Shallit, Predictable regular continued cotangent expansions, J. Res. Nat. Bur. Standards Sect. B 80B (1976), no. 2, 285-290.

For denominators see A006274.

Cf. A098316, A002814, A006268, A006271, A006275, A006276.

a := proc (n) option remember; if n = 1 then 10 else a(n-1)^3 + 3*a(n-1)^2 - 3 end if; end proc:
seq(a(n), n = 1..5); # Peter Bala, Jan 19 2022

From Peter Bala, Jan 19 2022: (Start)
a(n) = (11/2 + 3/2*sqrt(13))^3^(n-1) + (11/2 - 3/2*sqrt(13))^3^(n-1) - 1.
a(1) = 10 and a(n) = a(n-1)^3 + 3*a(n-1)^2 - 3 for n >= 2.
a(1) = 10 and a(n) = 13*(Product_{k = 1..n-1} a(k))^2 - 3 for n >= 2.
a(n) = A006268(n-1)^2 + 1 for n >= 1.
13 - 9*Product_{n = 1..N} (1 + 2/a(n))^2 = 52/(a(N+1) + 3). Therefore
sqrt(13) = 3*(1 + 2/10) * (1 + 2/1297) * (1 + 2/2186871697) * ... The convergence is cubic: the first six factors of the product give sqrt(13) correct to more than 750 decimal places.
3/sqrt(13) = (1 - 2/(10+2)) * (1 - 2/(1297+2)) * (1 - 2/(2186871697+2)) * .... (End)

A118242 Pierce expansion of 1/phi.

Original entry on oeis.org

1, 2, 4, 17, 19, 5777, 5779, 192900153617, 192900153619, 7177905237579946589743592924684177, 7177905237579946589743592924684179

Offset: 1

Eric W. Weisstein, Apr 17 2006

Differs from A006276 in the first term only.

G. C. Greubel, Table of n, a(n) for n = 1..17
Eric Weisstein's World of Mathematics, Pierce Expansion

Cf. A006276. A002813, A002814.

PierceExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@ NestList[{Floor[1/Expand[1 - #[[1]] #[[2]]]], Expand[1 - #[[1]] #[[2]]]} &, {Floor[1/(A - Floor[A])], A - Floor[A]}, n - 1]]; PierceExp[N[(Sqrt[5] - 1)/2, 7!], 10] (* G. C. Greubel, Nov 14 2016 *)

A118242(n) = A006276(n-1) for n>1.
From Peter Bala, Dec 03 2012: (Start)
Odd-indexed terms give A002813; even-indexed terms give A002814.
The Pierce series expansion is the alternating series 1/phi = 1/2*(sqrt(5) - 1) = 1/1 - 1/(1*2) + 1/(1*2*4) - 1/(1*2*4*17) + 1/(1*2*4*17*19) - ....
Another series expansion is
1/phi = a(1)/a(2) + (a(1)*a(3))/(a(2)*a(4)) + (a(1)*a(3)*a(5))/(a(2)*a(4)*a(6)) + ... = 1/2 + (1*4)/(2*17) + (1*4*19)/(2*17*5777) + ....
(End)

A006271 Numerators of a continued fraction for 1 + sqrt(2).

Original entry on oeis.org

2, 5, 197, 7761797, 467613464999866416197, 102249460387306384473056172738577521087843948916391508591105797

Offset: 0

N. J. A. Sloane

With b(n) = floor((1+sqrt(2))^n) (cf. A080039) the terms appear to be b(2*3^n). - Joerg Arndt, Apr 29 2013

Note that 1 + sqrt(2) = (c + sqrt(c^2+4))/2 and has regular continued fraction [c, c, ...] with c = 2. With b(n) = A006266(n), it can be expanded into an irregular continued fraction f(1) = b(1) and f(n) = (b[n-1]^2+1)/(b[n]-b[n-1]), and numerator(f(n)) = a(n) (cf. Shallit). - Michel Marcus, Apr 29 2013

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

F. L. Bauer, Letters to the editor: An Infinite Product for Square-Rooting with Cubic Convergence, The Mathematical Intelligencer, Vol. 20, Issue 1, (1998), 12-14.
N. J. Fine, Infinite products for k-th roots, Amer. Math. Monthly Vol. 84, No. 8 (Oct. 1977), pp. 629-630.
Jeffrey Shallit, Predictable regular continued cotangent expansions, J. Res. Nat. Bur. Standards Sect. B 80B (1976), no. 2, 285-290.

For denominators see A006272. Cf. A002814, A006266, A006273, A006275, A006276.

a := proc (n) option remember; if n = 1 then 5 else a(n-1)^3 + 3*a(n-1)^2 - 3 end if; end proc:
seq(a(n), n = 1 .. 5); # Peter Bala, Jan 19 2022

From Peter Bala, Jan 18 2022: (Start)
a(n) = (3 + 2*sqrt(2))^3^(n-1) + (3 - 2*sqrt(2))^3^(n-1) - 1 for n >= 1.
a(n) = A006266(n)^2 + 1 for n >= 1.
a(1) = 5 and a(n) = a(n-1)^3 + 3*a(n-1)^2 - 3 for n >= 2.
a(1) = 5 and a(n) = 8*(Product_{k = 1..n-1} a(k))^2 - 3 for n >= 2.
2 - Product_{n = 1..N} (1 + 2/a(n))^2 = 8/(a(N+1) + 3). Therefore
sqrt(2) = (1 + 2/5) * (1 + 2/197) * (1 + 2/7761797) * (1 + 2/ 467613464999866416197) * ... - see Bauer.
The convergence is cubic - see Fine. The first six factors of the product give sqrt(2) correct to more than 500 decimal places. (End)

Previous values for a(3) and a(4) were 776 and 1797. They have been merged into 7761797 to reflect the 2nd continued fraction on page 6 of Shallit paper by Michel Marcus, Apr 29 2013

A006270 Denominators of a continued fraction for (1 + sqrt(5))/2.

Original entry on oeis.org

1, 3, 72, 439128, 84722519069640072, 608130213374088941214747405817720857404971722895128

Offset: 0

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jeffrey Shallit, Predictable regular continued cotangent expansions, J. Res. Nat. Bur. Standards Sect. B 80B (1976), no. 2, 285-290.

For numerators see A002814.

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

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A006273 Numerators of a continued fraction for (3+sqrt(13))/2.

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A118242 Pierce expansion of 1/phi.

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A006271 Numerators of a continued fraction for 1 + sqrt(2).

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A006270 Denominators of a continued fraction for (1 + sqrt(5))/2.

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