%I #22 Jul 06 2019 15:26:56

%S 2,3,4,4,5,6,6,7,17,35,62,103,164,256,391,589,880,1303,1919,2814,4112,

%T 5993,8716,12655,18353,26591,38499,55710,80583,116523,168453,243485,

%U 351889,508506,734776,1061672,1533938,2216216

%N Floor of the solutions to c = exp(1 + n/c) for n >= 0, using recursion.

%C For n = 1 to 7 recursion produces convergence to single valued solutions.

%C For n >= 8 a dual-valued oscillating recursion persists between two stable values. The floor of the upper value for each n is included here. (The lower values of c are under 6 and approach exp(1) = 2.71828 for large n.)

%C At large n, the ratio of a(n)/a(n-1) approaches exp(1/exp(1)) = 1.444667861009 with more digits given by A073229.

%C At n = 0, c = exp(1).

%C At n = 1, c = 3.5911214766686 = A141251.

%C At n = 2, c = 4.3191365662914

%C At n = 3, c = 4.9706257595442

%C At n = 4, c = 5.5723925978776

%C At n = 5, c = 6.1383336446072

%C At n = 6, c = 6.6767832796664

%C At n = 7, c = 7.1932188286406

%C The convergence becomes "dual-valued" at n > exp(2) = 7.3890560989 = A072334.

%C At values of n = 7 and 8 the convergence is noticeably slower than at either larger or smaller values of n.

%C The recursion at n = exp(2) is only "quasi-stable" where c reluctantly approaches exp(2) = exp(1 + exp(2)/exp(2)) from any starting value, but never reaches it, and is not quite able to hold it if given the solution, due to machine rounding errors.

%F a(n) = floor(c) for the solutions to c = exp(1 + n/c) at n = 0 to 7, and the floor of the stable upper values of c for n >= 8.

%F Conjecture: a(n) = floor(e^(-e^(t^2/e^t - t)*t^2 + t + 1)) for all n > 13. - _Jon E. Schoenfield_, Jan 11 2014

%Y Cf. A073229, A141251.

%K nonn

%O 0,1

%A _Richard R. Forberg_, Dec 28 2013

%E Corrected and edited by _Jon E. Schoenfield_, Jan 11 2014