A059185 Engel expansion of Pi^2 = 9.8696...

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 9, 28, 45, 72, 111, 329, 415, 846, 1488, 5684, 1895742, 2890879, 5388452, 18083303, 30915293, 32699271, 38719784, 70637726, 118179186, 151342409, 995604288, 1839673662, 5342025157

Offset: 1

Author

Mitch Harris

Keywords

• nonn
• easy
• nice

Comments

Cf. A006784 for definition of Engel expansion.

References

• F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191.

Links

• G. C. Greubel and T. D. Noe, Table of n, a(n) for n = 1..1000 (terms 1 to 300 from T. D. Noe; terms 301 to 1000 from G. C. Greubel, Dec 27 2016)
• F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
• P. Erdős and Jeffrey Shallit, New bounds on the length of finite Pierce and Engel series, Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
• Index entries for sequences related to Engel expansions

Crossrefs

Cf. A002388 (Pi^2).

Programs

Mathematica

EngelExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@
NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]/1} &, {Ceiling[1/(A - Floor[A])], (A - Floor[A])/1}, n - 1]];
EngelExp[N[Pi^2, 7!], 100] (* modified by G. C. Greubel, Dec 27 2016 *)