A219506 Pierce expansion of 2 - sqrt(3).
3, 5, 51, 53, 140451, 140453, 2770663499604051, 2770663499604053, 21269209556953516583554114034636483645584976451, 21269209556953516583554114034636483645584976453
Offset: 0
Examples
We have the alternating series expansions 2 - sqrt(3) = 1/3 - 1/(3*5) + 1/(3*5*51) - 1/(3*5*51*53) + ... (2 - sqrt(3))^3 = 1/51 - 1/(51*53) + 1/(51*53*140451) - ... (2 - sqrt(3))^9 = 1/140451 - 1/(140451*140453) + ....
Links
- G. C. Greubel, Table of n, a(n) for n = 0..13
- 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.
- T. A. Pierce, On an algorithm and its use in approximating roots of algebraic equations, Amer. Math. Monthly, Vol. 36 No. 10, (1929) p.523-525.
- Jeffrey Shallit, Some predictable Pierce expansions, Fib. Quart., 22 (1984), 332-335.
- Eric Weisstein's World of Mathematics, Pierce Expansion
Programs
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Mathematica
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[2 - Sqrt[3] , 7!], 10] (* G. C. Greubel, Nov 14 2016 *)
Formula
a(2*n) = (2 + sqrt(3))^(3^n) + (2 - sqrt(3))^(3^n) - 1.
a(2*n + 1) = (2 + sqrt(3))^(3^n) + (2 - sqrt(3))^(3^n) + 1.
From Peter Bala, Jan 18 2022: (Start)
a(2*n+2) = a(2*n)^3 + 3*a(2*n)^2 - 3; a(2*n+1) = a(2*n-1)^3 - 3*a(2*n-1)^2 + 3.
a(2*n) = 6*(Product_{k = 1..n-1} a(2*k))^2 - 3, with a(0) = 1;
a(2*n+1) = 2*(Product_{k = 0..n-1} a(2*k+1))^2 + 3, with a(1) = 5.
sqrt(3) = (1 + 2/3)*(1 + 2/51)*(1 + 2/140451)*(1 + 2/2770663499604051)* .... See Bauer.
1/sqrt(3) = (1 - 2/5)*(1 - 2/53)*(1 - 2/140453)*(1 - 2/2770663499604053)* .... (End)
Comments