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A234604 Floor of the solutions to c = exp(1 + n/c) for n >= 0, using recursion.

%I A234604 #22 Jul 06 2019 15:26:56
%S A234604 2,3,4,4,5,6,6,7,17,35,62,103,164,256,391,589,880,1303,1919,2814,4112,
%T A234604 5993,8716,12655,18353,26591,38499,55710,80583,116523,168453,243485,
%U A234604 351889,508506,734776,1061672,1533938,2216216
%N A234604 Floor of the solutions to c = exp(1 + n/c) for n >= 0, using recursion.
%C A234604 For n = 1 to 7 recursion produces convergence to single valued solutions.
%C A234604 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 A234604 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 A234604 At n = 0, c = exp(1).
%C A234604 At n = 1, c = 3.5911214766686 = A141251.
%C A234604 At n = 2, c = 4.3191365662914
%C A234604 At n = 3, c = 4.9706257595442
%C A234604 At n = 4, c = 5.5723925978776
%C A234604 At n = 5, c = 6.1383336446072
%C A234604 At n = 6, c = 6.6767832796664
%C A234604 At n = 7, c = 7.1932188286406
%C A234604 The convergence becomes "dual-valued" at n > exp(2) = 7.3890560989 = A072334.
%C A234604 At values of n = 7 and 8 the convergence is noticeably slower than at either larger or smaller values of n.
%C A234604 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 A234604 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 A234604 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 A234604 Cf. A073229, A141251.
%K A234604 nonn
%O A234604 0,1
%A A234604 _Richard R. Forberg_, Dec 28 2013
%E A234604 Corrected and edited by _Jon E. Schoenfield_, Jan 11 2014