[seqfan] Further update on A112373

[Andrew N W Hone]

Dear Seqfans,

Using an exact formula for the terms of the sequence, that is a special case of Proposition 5 in the final version of my article (which has now appeared in Journal of Integer Sequences, Vol. 18 (2015), Article 15.8.4) one can show that the terms of A112373 are given by

x_n = smallest integer greater than or equal to X_n,

that is, the ceiling of X_n, where

X_n = \exp ( A k^n + B k^{-n} ),

with A = 0.1468643109..., B = -2.719173202... and k = (3+\sqrt{5})/2.

The exact formulae for A and B are

A = ( k - k^{-1} )^{-1} \sum_{j=1}^{\infty} k^{-j} \log ( 1 + 1/x_j ),

B = -( k - k^{-1} )^{-1} \sum_{j=1}^{\infty} k^{j} \log ( 1 + 1/x_j ),

so one requires the terms of the sequence to compute them! Nevertheless, if enough decimal digits are used, it seems that with the approximation to A and B obtained by truncating each of the sums at the nth term, one can correctly compute x_{n+1} from the ceiling of X_{n+1}.

The above results are all based on the method described in

A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quart. (1973), 429-437.

Best wishes,
Andy