An n th-Cousin Mating Model and the n -Anacci Numbers

In seeking to understand the size of inbred pedigrees, J. Lachance (J. Theor. Biol. 261, 238–247, 2009) studied a population model in which, for a fixed value of n, each mating occurs between nth cousins. We explain a connection between the second-cousin case of the model (n=2) and the Fibonacci sequence, and more generally, between the nth-cousin case and the n-anacci sequence (n≥2). For a model with nth-cousin mating (n≥1), we obtain the generating function describing the size of the pedigree t generations back from the present—tabulating the numbers of lineal genealogical ancestors in specified generations—and we use it to evaluate the asymptotic growth of the pedigree size. In particular, we show that the pedigree growth rate asymptotically follows the growth rate of the n-anacci sequence, and in particular, the golden ratio ϕ=(1+5)/2≈1.61803 for second-cousin mating n=2, the tribonacci growth constant 1.83928… for third-cousin mating n=3, and the tetranacci growth constant 1.92756… for fourth-cousin mating n=4. The growth rate approaches 2 as n increases. The computations explain the appearance of familiar numerical sequences and constants in a pedigree model. They also recall similar appearances of such sequences and constants in studies of population biology more generally.