A118242 Pierce expansion of 1/phi.

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

Offset: 1

Author

Eric W. Weisstein, Apr 17 2006

Keywords

• nonn
• easy

Comments

Differs from A006276 in the first term only.

Links

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

Crossrefs

Cf. A006276. A002813, A002814.

Programs

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[(Sqrt[5] - 1)/2, 7!], 10] (* G. C. Greubel, Nov 14 2016 *)

Formula

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)