TITLE: If $n$ birds are sitting in circle, each pecks its left or right bird with equal probability. What is the distribution of number of pecked birds?
QUESTION [5 upvotes]: If n birds are sitting in circle, each pecks its left or right bird with equal probability. 
What is the distribution of number of pecked birds?
Note at least $\frac{n}{2}$ birds get pecked, so it cannot be binomial. 
Also note that analysis for number of unpecked birds is $\frac{n}{4}$ for $n > 2$, which is actually just a coincidence if you assume binomial. For $n=1$ and $n=2$ all birds are necessarily pecked. 
Also if there are $5$ birds a,b,c,d,e in order, if c is not pecked then both a and e must be pecked, so treating them as independent events are also not correct. 
The distribution looks like this for various values of $n$. (Divide by $2^n$ as to get probability)
1   {1 -> 2}
2   {2 -> 4}
3   {2 -> 6, 3 -> 2}
4   {2 -> 4, 3 -> 8, 4 -> 4}
5   {3 -> 10, 4 -> 20, 5 -> 2}
6   {4 -> 36, 5 -> 24, 6 -> 4}
7   {4 -> 14, 5 -> 70, 6 -> 42, 7 -> 2}
8   {4 -> 4, 5 -> 48, 6 -> 152, 7 -> 48, 8 -> 4}
9   {5 -> 18, 6 -> 168, 7 -> 252, 8 -> 72, 9 -> 2}
10  {6 -> 100, 7 -> 400, 8 -> 440, 9 -> 80, 10 -> 4}
11  {6 -> 22, 7 -> 330, 8 -> 924, 9 -> 660, 10 -> 110, 11 -> 2}
12  {6 -> 4, 7 -> 120, 8 -> 1020, 9 -> 1808, 10 -> 1020, 11 -> 120, 12 -> 4}
13  {7 -> 26, 8 -> 572, 9 -> 2574, 10 -> 3432, 11 -> 1430, 12 -> 156, 13 -> 2}
14  {8 -> 196, 9 -> 1960, 10 -> 6076, 11 -> 5936, 12 -> 2044, 13 -> 168, 14 -> 4}
15  {8 -> 30, 9 -> 910, 10 -> 6006, 11 -> 12870, 12 -> 10010, 13 -> 2730, 14 -> 210, 15 -> 2}
16  {8 -> 4, 9 -> 224, 10 -> 3696, 11 -> 15904, 12 -> 25880, 13 -> 15904, 14 -> 3696, 15 -> 224, 16 -> 4}
17  {9 -> 34, 10 -> 1360, 11 -> 12376, 12 -> 38896, 13 -> 48620, 14 -> 24752, 15 -> 4760, 16 -> 272, 17 -> 2}
18  {10 -> 324, 11 -> 6048, 12 -> 37296, 13 -> 87264, 14 -> 87768, 15 -> 36960, 16 -> 6192, 17 -> 288, 18 -> 4}
19  {10 -> 38, 11 -> 1938, 12 -> 23256, 13 -> 100776, 14 -> 184756, 15 -> 151164, 16 -> 54264, 17 -> 7752, 18 -> 342, 19 -> 2}
20  {10 -> 4, 11 -> 360, 12 -> 9780, 13 -> 77280, 14 -> 252360, 15 -> 369008, 16 -> 252360, 17 -> 77280, 18 -> 9780, 19 -> 360, 20 -> 4}
21  {11 -> 42, 12 -> 2660, 13 -> 40698, 14 -> 232560, 15 -> 587860, 16 -> 705432, 17 -> 406980, 18 -> 108528, 19 -> 11970, 20 -> 420, 21 -> 2}
22  {12 -> 484, 13 -> 14520, 14 -> 149556, 15 -> 638880, 16 -> 1294216, 17 -> 1292368, 18 -> 640200, 19 -> 148896, 20 -> 14740, 21 -> 440, 22 -> 4}
23  {12 -> 46, 13 -> 3542, 14 -> 67298, 15 -> 490314, 16 -> 1634380, 17 -> 2704156, 18 -> 2288132, 19 -> 980628, 20 -> 201894, 21 -> 17710, 22 -> 506, 23 -> 2}
24  {12 -> 4, 13 -> 528, 14 -> 21384, 15 -> 268752, 16 -> 1471932, 17 -> 3920928, 18 -> 5410160, 19 -> 3920928, 20 -> 1471932, 21 -> 268752, 22 -> 21384, 23 -> 528, 24 -> 4}
25  {13 -> 50, 14 -> 4600, 15 -> 106260, 16 -> 961400, 17 -> 4085950, 18 -> 8914800, 19 -> 10400600, 20 -> 6537520, 21 -> 2163150, 22 -> 354200, 23 -> 25300, 24 -> 600, 25 -> 2}
26  {14 -> 676, 15 -> 29744, 16 -> 461032, 17 -> 3123120, 18 -> 10626044, 19 -> 19311968, 20 -> 19318832, 21 -> 10620896, 22 -> 3125980, 23 -> 459888, 24 -> 30056, 25 -> 624, 26 -> 4}
27  {14 -> 54, 15 -> 5850, 16 -> 161460, 17 -> 1776060, 18 -> 9373650, 19 -> 26075790, 20 -> 40116600, 21 -> 34767720, 22 -> 16872570, 23 -> 4440150, 24 -> 592020, 25 -> 35100, 26 -> 702, 27 -> 2}
28  {14 -> 4, 15 -> 728, 16 -> 41132, 17 -> 752752, 18 -> 6218212, 19 -> 26242216, 20 -> 60849516, 21 -> 80226336, 22 -> 60849516, 23 -> 26242216, 24 -> 6218212, 25 -> 752752, 26 -> 41132, 27 -> 728, 28 -> 4}
29  {15 -> 58, 16 -> 7308, 17 -> 237510, 18 -> 3121560, 19 -> 20030010, 20 -> 69194580, 21 -> 135727830, 22 -> 155117520, 23 -> 103791870, 24 -> 40060020, 25 -> 8584290, 26 -> 950040, 27 -> 47502, 28 -> 812, 29 -> 2}

REPLY [2 votes]: Assume that the number $n$ of birds is odd. The behavior of the birds is encoded in a circular word $w$ of length $n$ over the alphabet $\{L,R\}$. A particular bird is not pecked iff in the neighborhood of that bird the word looks like $\ldots L*R\ldots$, whereby the $*$ marks the letter of that  bird. We therefore rearrange the word $w$ such that the numbers of the birds appear in the new (circular) order
$$1,\ 3,\ 5,\ \ldots,\ n,\ 2,\ 4,\ \ldots,\ (n-1)\ .$$
Let $w'$ be the resulting word. The number of unpecked birds then is equal to the number of $LR$-subwords in $w'$.
There are $2^n$ possible words $w'$, all of them equiprobable. How many of them have exactly $k$ subwords $LR$? If there are $k$ subwords $LR$ then there have to be $k$ subwords $RL$ as well, and the two types alternate. We can choose the $2k$ jump spots in ${n\choose 2k}$ ways, and then  identify the $LR$-jumps in $2$ ways. It follows that the probability $p(k)$ for exactly $k$ unpecked birds is given by
$$p(k)={2\over2^n}{n\choose2k}\qquad\bigl(0\leq k\leq\lfloor n/2\rfloor\bigr)\ .\tag{1}$$
Note:  $\ 2\sum_{k\geq0}{n\choose 2k}=(1+1)^n+(1-1)^n=2^n$. – I checked a few cases of $(1)$. The results corroborated your data.
If $n=2m$ is even one has to split the word $w$ into two circular words $w'$ and $w''$ of length $m$ each. The final formula for $p(k)$ then involves a convolution of binomial coefficients, which can perhaps be simplified.