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What is the asymptotic growth of the sequence $$a_n:=\sum_{k\geq 0} 3^k c_{n,k},$$ as $n\rightarrow\infty$, where $c_{n,k}$ denotes the number of integer compositions of $n$ with exactly $k$ many 2s?

A composition of $n$ is a sum $n=c_1+c_2+\cdots+c_p$, with all the $c_i$ positive. The first values of the sequence $a_n$ are $1,1,4,8,22,52,135,\ldots$ (not in the OEIS). [Edit: As pointed out by Somos below, the value 135 is wrong, and must be corrected to 132, and then the sequence is in the OEIS.]

So far, I was only able to prove the following bounds: As $\sum_{k\geq 0}c_{n,k}=2^{n-1}$, it follows that $$2^{n-1}\leq a_n \leq (2\sqrt{3})^n=(3.464...)^n.$$

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  • $\begingroup$ Can we say $c_{n,k}=c_{n-2,k-1}k+1)$, and thus produce a recursion? $\endgroup$
    – vidyarthi
    Jun 21 '19 at 17:05
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    $\begingroup$ $c_{n,k}$ is the sequence A105114 in the OEIS. $\endgroup$ Jun 21 '19 at 21:56
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When corrected, $a_n$ is the OEIS sequence A052528 whose first nine values are $\,1,1,4,8,22,52,132,324,808,\dots\,$ and it has a linear recurrence.

The combinatorial recurrence from its combination definition leads immediately to $$ a_n = 2a_{n-2} + \sum_{k=0}^{n-1} a_k $$ where the $\, 2a_{n-2}\,$ comes from the combination part $2.$ The ordinary generating function is $$ (1 - x)/(1 - 2 x - 2 x^2 + 2 x^3). $$ The growth rate of $\,a_n\,$ depends on the reciprocal of the smallest root $\,\alpha\,$ of $\, 2x^3 -2x^2 -2x +1.$ Thus $\,1/\alpha \approx 2.4811943\,$ so that $\,a_n \propto 1/\alpha^n.$

EDIT: The sequence $\,c_{n,k}\,$ is the triangular OEIS sequence A105114

Triangle read by rows: T(n,k) is the number of compositions of n having exactly k parts equal to 2.

Again, its combination definition leads immediately to $$ c_{n,k} = \sum_{j=1}^n c_{n-j,k-[j=2]} $$ where $\,[j=2] := 1\,$ if $j=2$, else $0$.

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    $\begingroup$ I think there is also a multiplied constant, the limit of $(1-x/\alpha)\,\mathrm{gf}(x)$ as $x\to\alpha$, which is about 0.56158611. $\endgroup$ Jun 21 '19 at 19:05
  • $\begingroup$ @BrendanMcKay Yes, I really meant $\propto$ instead of $\sim$. Thanks for your comment. $\endgroup$
    – Somos
    Jun 21 '19 at 19:17
  • $\begingroup$ Wonderful, thank you! The one mistake in computations I made with 135 instead of 132 spoiled my OEIS search. After entering 1,1,4,8,22,52, there were still several sequences matching, and then after entering 135, none was left. $\endgroup$ Jun 22 '19 at 16:12

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