

A234604


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


1



2, 3, 4, 4, 5, 6, 6, 7, 17, 35, 62, 103, 164, 256, 391, 589, 880, 1303, 1919, 2814, 4112, 5993, 8716, 12655, 18353, 26591, 38499, 55710, 80583, 116523, 168453, 243485, 351889, 508506, 734776, 1061672, 1533938, 2216216
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OFFSET

0,1


COMMENTS

For n = 1 to 7 recursion produces convergence to single valued solutions.
For n >= 8 a dualvalued 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.)
At large n, the ratio of a(n)/a(n1) approaches exp(1/exp(1)) = 1.444667861009 with more digits given by A073229.
At n = 0, c = exp(1).
At n = 1, c = 3.5911214766686 = A141251.
At n = 2, c = 4.3191365662914
At n = 3, c = 4.9706257595442
At n = 4, c = 5.5723925978776
At n = 5, c = 6.1383336446072
At n = 6, c = 6.6767832796664
At n = 7, c = 7.1932188286406
The convergence becomes "dualvalued" at n > exp(2) = 7.3890560989 = A072334.
At values of n = 7 and 8 the convergence is noticeably slower than at either larger or smaller values of n.
The recursion at n = exp(2) is only "quasistable" 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.


LINKS

Table of n, a(n) for n=0..37.


FORMULA

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.
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


CROSSREFS

Cf. A073229, A141251.
Sequence in context: A095769 A080820 A274687 * A236346 A306890 A116549
Adjacent sequences: A234601 A234602 A234603 * A234605 A234606 A234607


KEYWORD

nonn


AUTHOR

Richard R. Forberg, Dec 28 2013


EXTENSIONS

Corrected and edited by Jon E. Schoenfield, Jan 11 2014


STATUS

approved



