A059185 - OEIS

%I #27 Sep 02 2023 18:54:12

%S 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,

%T 1895742,2890879,5388452,18083303,30915293,32699271,38719784,70637726,

%U 118179186,151342409,995604288,1839673662,5342025157

%N Engel expansion of Pi^2 = 9.8696...

%C Cf. A006784 for definition of Engel expansion.

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

%H G. C. Greubel and T. D. Noe, <a href="/A059185/b059185.txt">Table of n, a(n) for n = 1..1000</a> (terms 1 to 300 from T. D. Noe; terms 301 to 1000 from G. C. Greubel, Dec 27 2016)

%H F. Engel, <a href="/A006784/a006784.pdf">Entwicklung der Zahlen nach Stammbruechen</a>, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.

%H P. Erdős and Jeffrey Shallit, <a href="http://www.numdam.org/item?id=JTNB_1991__3_1_43_0">New bounds on the length of finite Pierce and Engel series</a>, Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.

%H <a href="/index/El#Engel">Index entries for sequences related to Engel expansions</a>

%t EngelExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@

%t NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]/1} &, {Ceiling[1/(A - Floor[A])], (A - Floor[A])/1}, n - 1]];

%t EngelExp[N[Pi^2, 7!], 100] (* modified by _G. C. Greubel_, Dec 27 2016 *)

%Y Cf. A002388 (Pi^2).

%K nonn,easy,nice

%O 1,10

%A _Mitch Harris_