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

Author

G. C. Greubel, Dec 25 2016

Keywords

• nonn

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.

Examples

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..250
• Eric Weisstein's World of Mathematics, Pierce Expansion
• Wikipedia, Engel Expansion

Crossrefs

Cf. A232325.

Programs

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]