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A similar idea was considered by Motoo Kimura and Tomoko Ohta in The Average Number of Generations Until Fixation of a Mutant Gene. They used a diffusion model, which should correspond to looking at the behavior as $N \rightarrow \infty$. The relevant line should be:

NOW, in a population consisting of N individuals, if we assume that each mutant gene is represented only once at the moment of its occurrence, p = 1/ (2N), and from formula (14), the average number of generations until fixation of a neutral mutation becomes $\bar{t}_1(\frac{1}{2N}) = -8 N N_e (1 - \frac{1}{2N}) \log_e(1 - \frac{1}{2N})$

Our $N$ is Kimura and Crow's $2N$; if I understand correctly, their $N_e$ should equal $N$ in our situation ($N_e$ denotes an "effective" population size where individuals act differently; here, that isn't true). Correspondingly, our limiting behavior should be:

$-2 N^2 (1 - \frac{1}{N}) \ln(1 - \frac{1}{N}) = 2 N (N - 1) \sum_{i = 1}^\infty \frac{1}{i N^i} = 2 \left(\sum_{i = 1}^\infty \frac{1}{i N^{i - 2}} - \sum_{i = 1}^\infty\frac{1}{i N^{i - 1}}\right)$

$= 2N + 2\sum_{i = 0}^\infty \frac{1}{N^i} (-\frac{1}{i + 1} + \frac{1}{i + 2}) = 2N - 2\sum_{i = 0}^\infty \frac{1}{(i + 1)(i + 2) N^i}$

$ = 2N - 1 - \frac{1}{3N} - O(\frac{1}{N^2})$

This is subject to the diffusion model being accurate to this degree of approximation, which I have no specific support for. I wouldn't be surprised to see the logarithmic term in Angela Zhou's answer show up when considering the effect of the first "round".

Tracing back the references a bit, I found this paper, which may be more obviously relevant (as well as already being written for mathematicians directly).

A similar idea was considered by Motoo Kimura and Tomoko Ohta in The Average Number of Generations Until Fixation of a Mutant Gene. They used a diffusion model, which should correspond to looking at the behavior as $N \rightarrow \infty$. The relevant line should be:

NOW, in a population consisting of N individuals, if we assume that each mutant gene is represented only once at the moment of its occurrence, p = 1/ (2N), and from formula (14), the average number of generations until fixation of a neutral mutation becomes $\bar{t}_1(\frac{1}{2N}) = -8 N N_e (1 - \frac{1}{2N}) \log_e(1 - \frac{1}{2N})$

Our $N$ is Kimura and Ohta's $2N$; if I understand correctly, their $N_e$ should equal $N$ in our situation ($N_e$ denotes an "effective" population size where individuals act differently; here, that isn't true). Correspondingly, our limiting behavior should be:

$-2 N^2 (1 - \frac{1}{N}) \ln(1 - \frac{1}{N}) = 2 N (N - 1) \sum_{i = 1}^\infty \frac{1}{i N^i} = 2 \left(\sum_{i = 1}^\infty \frac{1}{i N^{i - 2}} - \sum_{i = 1}^\infty\frac{1}{i N^{i - 1}}\right)$

$= 2N + 2\sum_{i = 0}^\infty \frac{1}{N^i} (-\frac{1}{i + 1} + \frac{1}{i + 2}) = 2N - 2\sum_{i = 0}^\infty \frac{1}{(i + 1)(i + 2) N^i}$

$ = 2N - 1 - \frac{1}{3N} - O(\frac{1}{N^2})$

This is subject to the diffusion model being accurate to this degree of approximation, which I have no specific support for. I wouldn't be surprised to see the logarithmic term in Angela Zhou's answer show up when considering the effect of the first "round".

Tracing back the references a bit, I found this paper, which may be more obviously relevant (as well as already being written for mathematicians directly).

A similar idea was considered by Motoo Kimura and Tomoko Ohta in The Average Number of Generations Until Fixation of a Mutant Gene. They used a diffusion model, which should correspond to looking at the behavior as $N \rightarrow \infty$. The relevant line should be:

NOW, in a population consisting of N individuals, if we assume that each mutant gene is represented only once at the moment of its occurrence, p = 1/ (2N), and from formula (14), the average number of generations until fixation of a neutral mutation becomes $\bar{t}_1(\frac{1}{2N}) = -8 N N_e (1 - \frac{1}{2N}) \log_e(1 - \frac{1}{2N})$

Our $N$ is Kimura and Crow's $2N$; if I understand correctly, their $N_e$ should equal $N$ in our situation ($N_e$ denotes an "effective" population size where individuals act differently; here, that isn't true). Correspondingly, our limiting behavior should be:

$-2 N^2 (1 - \frac{1}{N}) \ln(1 - \frac{1}{N}) = 2 N (N - 1) \sum_{i = 1}^\infty \frac{1}{i N^i} = 2 \left(\sum_{i = 1}^\infty \frac{1}{i N^{i - 2}} - \sum_{i = 1}^\infty\frac{1}{i N^{i - 1}}\right)$

$= 2N + 2\sum_{i = 0}^\infty \frac{1}{N^i} (-\frac{1}{i + 1} + \frac{1}{i + 2}) = 2N - 2\sum_{i = 0}^\infty \frac{1}{(i + 1)(i + 2) N^i}$

$ = 2N - 1 - \frac{1}{3N} - O(\frac{1}{N^2})$

This is subject to the diffusion model being accurate to this degree of approximation, which I have no specific support for. I wouldn't be surprised to see the logarithmic term in Angela Zhou's answer show up when considering the effect of the first "round".

A similar idea was considered by Motoo Kimura and Tomoko Ohta in The Average Number of Generations Until Fixation of a Mutant Gene. They used a diffusion model, which should correspond to looking at the behavior as $N \rightarrow \infty$. The relevant line should be:

NOW, in a population consisting of N individuals, if we assume that each mutant gene is represented only once at the moment of its occurrence, p = 1/ (2N), and from formula (14), the average number of generations until fixation of a neutral mutation becomes $\bar{t}_1(\frac{1}{2N}) = -8 N N_e (1 - \frac{1}{2N}) \log_e(1 - \frac{1}{2N})$

Our $N$ is Kimura and Crow's $2N$; if I understand correctly, their $N_e$ should equal $N$ in our situation ($N_e$ denotes an "effective" population size where individuals act differently; here, that isn't true). Correspondingly, our limiting behavior should be:

$-2 N^2 (1 - \frac{1}{N}) \ln(1 - \frac{1}{N}) = 2 N (N - 1) \sum_{i = 1}^\infty \frac{1}{i N^i} = 2 \left(\sum_{i = 1}^\infty \frac{1}{i N^{i - 2}} - \sum_{i = 1}^\infty\frac{1}{i N^{i - 1}}\right)$

$= 2N + 2\sum_{i = 0}^\infty \frac{1}{N^i} (-\frac{1}{i + 1} + \frac{1}{i + 2}) = 2N - 2\sum_{i = 0}^\infty \frac{1}{(i + 1)(i + 2) N^i}$

$ = 2N - 1 - \frac{1}{3N} - O(\frac{1}{N^2})$

This is subject to the diffusion model being accurate to this degree of approximation, which I have no specific support for. I wouldn't be surprised to see the logarithmic term in Angela Zhou's answer show up when considering the effect of the first "round".

Tracing back the references a bit, I found this paper, which may be more obviously relevant (as well as already being written for mathematicians directly).

A similar idea was considered by Motoo Kimura and Tomoko Ohta in The Average Number of Generations Until Fixation of a Mutant Gene. They used a diffusion model, which should correspond to looking at the behavior as $N \rightarrow \infty$. The relevant line should be:

NOW, in a population consisting of N individuals, if we assume that each mutant gene is represented only once at the moment of its occurrence, p = 1/ (2N), and from formula (14), the average number of generations until fixation of a neutral mutation becomes $\bar{t}_1(\frac{1}{2N}) = -8 N N_e (1 - \frac{1}{2N}) \log_e(1 - \frac{1}{2N})$

Our $N$ is Kimura and Crow's $2N$; if I understand correctly, their $N_e$ should equal $N$ in our situation ($N_e$ denotes an "effective" population size where individuals act differently; here, that isn't true). Correspondingly, our limiting behavior should be:

$-2 N^2 (1 - \frac{1}{N}) \ln(1 - \frac{1}{N}) = 2 N (N - 1) \sum_{i = 1}^\infty \frac{1}{i N^i} = 2 \left(\sum_{i = 1}^\infty \frac{1}{i N^{i - 2}} - \sum_{i = 1}^\infty\frac{1}{i N^{i - 1}}\right)$

$= 2N + 2\sum_{i = 0}^\infty \frac{1}{N^i} (-\frac{1}{i + 1} + \frac{1}{i + 2}) = 2N - 2\sum_{i = 0}^\infty \frac{1}{(i + 1)(i + 2) N^i}$

This is subject to the diffusion model being accurate to this degree of approximation, which I have no specific support for. I wouldn't be surprised to see the logarithmic term in Angela Zhou's answer show up when considering the effect of the first "round".

A similar idea was considered by Motoo Kimura and Tomoko Ohta in The Average Number of Generations Until Fixation of a Mutant Gene. They used a diffusion model, which should correspond to looking at the behavior as $N \rightarrow \infty$. The relevant line should be:

NOW, in a population consisting of N individuals, if we assume that each mutant gene is represented only once at the moment of its occurrence, p = 1/ (2N), and from formula (14), the average number of generations until fixation of a neutral mutation becomes $\bar{t}_1(\frac{1}{2N}) = -8 N N_e (1 - \frac{1}{2N}) \log_e(1 - \frac{1}{2N})$

Our $N$ is Kimura and Crow's $2N$; if I understand correctly, their $N_e$ should equal $N$ in our situation ($N_e$ denotes an "effective" population size where individuals act differently; here, that isn't true). Correspondingly, our limiting behavior should be:

$-2 N^2 (1 - \frac{1}{N}) \ln(1 - \frac{1}{N}) = 2 N (N - 1) \sum_{i = 1}^\infty \frac{1}{i N^i} = 2 \left(\sum_{i = 1}^\infty \frac{1}{i N^{i - 2}} - \sum_{i = 1}^\infty\frac{1}{i N^{i - 1}}\right)$

$= 2N + 2\sum_{i = 0}^\infty \frac{1}{N^i} (-\frac{1}{i + 1} + \frac{1}{i + 2}) = 2N - 2\sum_{i = 0}^\infty \frac{1}{(i + 1)(i + 2) N^i}$

$ = 2N - 1 - \frac{1}{3N} - O(\frac{1}{N^2})$

This is subject to the diffusion model being accurate to this degree of approximation, which I have no specific support for. I wouldn't be surprised to see the logarithmic term in Angela Zhou's answer show up when considering the effect of the first "round".