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A118242
Pierce expansion of 1/phi.
3
1, 2, 4, 17, 19, 5777, 5779, 192900153617, 192900153619, 7177905237579946589743592924684177, 7177905237579946589743592924684179
OFFSET
1,2
COMMENTS
Differs from A006276 in the first term only.
LINKS
Eric Weisstein's World of Mathematics, Pierce Expansion
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)
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 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Eric W. Weisstein, Apr 17 2006
STATUS
approved