Number of Relatives
19.09.2022
Anyone who is involved in genealogy may have asked themselves how many ancestors there are or how many people you are related to. If one assumes that every human being has two physical parents (meanwhile, there are children who have 3 parents, but this case is not considered here) then the calculation of the direct ancestors is quite simple. Each person has 2 parents, 4 grandparents, 8 great-grandparents, etc. That means there are 2^n ancestors in the n-th generation.
Charlemagne was born around the year 748 - that is 1274 years ago. Genealogists put an average of 30 years on a generation, so he lived more than 42 generations ago. If one looks back until this time each of us would have 2^42 ancestors - an unimaginably large number, which goes into trillions. However, in Central Europe lived at this time no more than 50 million people. How can that be?
The reason therefore is called Pedigree collapse. This typically happens if relatives have children together. The descendants have then less ancestors than the maximum possible number.
Calculating the number of genetic relative people
In our theoretical calculation of the number of all genetic relative people up to a past generation n we assume idealized conceptions: On the one hand, we ignore the loss of ancestors, on the other hand we assume that each pair of parents has exactly k children.
Let's consider the example with k=2
Generation 1: 2 parents (G1) + 2 children (G0) = 4 relatives
Generation 2: 4 grandparents (G2) + 4 children (G1) + 6 children (G0) = 14 relatives
Generation 3: 8 great-grandparents (G3) + 8 children (G2) + 12 children (G1) + 22 children (G0) = 50
relatives
In the 10th generation this would be 2^10 = 1024 ancestors over which one would be genetic related to
700074 people.
Another example with k=3
The person themself as well as the direct ancestors are shown in black, the persons with which one is also
genetically related in red. If each couple has 3 children, then in addition to the direct ancestors
there are also 2 siblings, 4 uncles or aunts and 12 cousins. Thus a total of 25 genetically related persons
are represented in this genealogical table with a generation depth of n=2.
Mathematical formula
Without regarding the formula in more detail, the number of all relatives for n generations and k children
(k >= 2) can be calculated by the following sum formula:
R(n,k) = Sum[i=0..n]2^n + Sum[i=1..n]2^(i-1)*((k^i)-1)
This sum can also be calculated directly using the following formula:
R(n,k) = 2^n+(k*(2*k)^n-k)/(2*k-1)
Number series
For different numbers of children k, the following series of numbers result for generations 0 to 10:
k=1: 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047 (The person themself plus the number of direct ancestors)
k=2: 1, 4, 14, 50, 186, 714, 2794, 11050, 43946, 175274, 700074
k=3: 1, 5, 25, 137, 793, 4697, 28057, 168089, 1008025, 6047129, 36280729
k=4: 1, 6, 40, 300, 2356, 18756, 149860, 1198500, 9587236, 76696356, 613567780
k=5: 1, 7, 59, 563, 5571, 55587, 555619, 5555683, 55555811, 555556067, 5555556579
k=6: 1, 8, 82, 950, 11326, 135758, 1628782, 19544750, 234535726, 2814426158, 33773108782
k=7: 1, 9, 109, 1485, 20701, 289629, 4054429, 56761245, 794655901, 11125179549, 155752507549
Explicit formulas
k=1: a(n) = 2^n + 1*( 2^n - 1)/ 1
k=2: a(n) = 2^n + 2*( 4^n - 1)/ 3
k=3: a(n) = 2^n + 3*( 6^n - 1)/ 5
k=4: a(n) = 2^n + 4*( 8^n - 1)/ 7
k=5: a(n) = 2^n + 5*(10^n - 1)/ 9
k=6: a(n) = 2^n + 6*(12^n - 1)/11
k=7: a(n) = 2^n + 7*(14^n - 1)/13
Program in Python to generate the number series
for k in range(1,8):
l=[];
for n in range(0,11):
v = 2**n+(k*(2*k)**n-k)//(2*k-1)
l.append(int(v))
print(l)
Interesting
Reagarding the case k=5 minus the number of ancestors of the the n-th generation (n >= 1), we get the number series 5, 55, 555, 5555, 55555, 555555, 5555555, 55555555, 555555555, 5555555555, ...
References
k=1: OEIS Serie A000225
k=2: OEIS Serie A076024
k=3: OEIS Serie A358504
k=4: OEIS Serie A358598
k=5: OEIS Serie A358599
k=6: OEIS Serie A358600
k=7: OEIS Serie A358601
See also
OEIS Serie A000079
OEIS Serie A002279