A182503 Engel expansion of the Dottie number, A003957.

2, 3, 3, 4, 5, 15, 17, 66, 196, 233, 284, 375, 1613, 2131, 3574, 14122, 24171, 49097, 56871, 69361, 193406, 243145, 289951, 682749, 14501588, 21191773, 121635191, 810759781, 1292785381, 136110231377, 294401497761

Offset: 1

Author

Ben Branman, May 02 2012

Keywords

• nonn
• easy

Comments

Dottie number = 1/2 + 1/2/3 + 1/2/3/3 + 1/2/3/3/4 + 1/2/3/3/4/5 + 1/2/3/3/4/5/15 +...

Links

• G. C. Greubel, Table of n, a(n) for n = 1..500
• Jean-Christophe Pain, An exact series expansion for the Dottie number, arXiv:2303.17962 [math.NT], 2023.
• Eric Weisstein's World of Mathematics, Engel Expansion
• Eric Weisstein's World of Mathematics, Dottie Number
• Index entries for sequences related to Engel expansions

Crossrefs

Cf. A003957, A006784.

Programs

Mathematica

EngelExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]} &, {Ceiling[1/(A - Floor[A])], A - Floor[A]}, n - 1]]; z = FindRoot[x == Cos[x], {x, 1}, WorkingPrecision -> 10000][[1, -1]]; EngelExp[z, 30]