A280095 Engel expansion of phi to the base Pi.

2, 105, 617, 3077, 9757, 71731, 306407, 2071853, 10770894, 185768753, 1672941615, 14465494561, 338610760068, 1260607468485, 5168248479349, 151720540392580, 1384591426590643, 30464122079618738, 121074568909128689, 574695040334652831

Offset

0,1

Comments

The Mathematica code provided calculates (1+ sqrt(5))/4 and yields the Engel expansion (1+sqrt(5))/4 = Pi/4 + Pi^2/(4*105) + O(Pi^6). Multiplying this expansion by 2 gives this sequence.

Links

G. C. Greubel, Table of n, a(n) for n = 0..250

Eric Weisstein's World of Mathematics, Pierce Expansion

Wikipedia, Engel Expansion

Example

phi = Pi/2 + Pi^2/(2*105) + Pi^3/(2*105*617) + ...

Mathematica

EngelExp[A_, n_] := Join[Array[Pi &, Floor[A]], First@Transpose@
NestList[{Ceiling[Pi/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]/Pi} &, {Ceiling[Pi/(A - Floor[A])], (A - Floor[A])/Pi}, n - 1]]; EngelExp[N[(1 + Sqrt[5])/4, 7!], 20]

Crossrefs

Cf. A232325.

Sequence in context: A042351 A258828 A156880 * A229016 A222840 A356723

Adjacent sequences: A280092 A280093 A280094 * A280096 A280097 A280098

Keyword

nonn

Author

G. C. Greubel, Dec 25 2016

Status

approved