Let us denote the cardinality of the subsets of {1,..,n} without two consequent numbers

(e.g. {..,4,5,..} is not allowed) by A[n]. What is the asymptotics of A[n] as n approaches infinity?

The same question for the case of three consequent numbers.

Here is my math experiment.

restart; L := combinat:-powerset({seq(i, i = 1 .. 11)}):#n = 11

nops(%);

2048

M := selectremove(c-> min([seq(c[k+1]-c[k], k = 1 .. nops(c)-1)]) = 1, L)[2]:

nops(M);

233

The other results are [11, 233], [15, 1597], [20, 17711], [21, 28657], [22, 46368].

These points are very close to some straight line in logarithmic scale as

plot([[11, 233], [15, 1597], [20, 17711], [21, 28657], [22, 46368]], axis[2] = [mode = log]);

shows. However, the ones do not exactly belong to a straight line:

evalf(ln(46368)-ln(28657), 15);

0.4812118247230

evalf(ln(28657)-ln(17711), 15);

0.48121182594077

eval(exp(.4812118247230*n), n = 15);

1364.000725 .

These results suggest that A[n] is asymptotically equal to exp(c*n) with c about 0.481.

I have not succeeded to find out the nature of the constant c.

question_on_asymptotics.mw