[seqfan] 2nd email; "Clonacci numbers, from interpretation of Brandon, R., 1990, Adaptation and Environment. Princeton: Princeton University Press."

[Jonathan Post]

[2nd, shorter email, with no attachment, per seqfan administrator
request.  Formatted Word file may be sent to anyone who requests it]

Suppose we have asexual immortal creatures, call them orgs, which are
alternately very fertile, able to give birth to three clones at the
very end of the year when they become age 1, 3, 5, 7, and all other
odd ages; and slightly fertile, able to give birth to a single new
clone at age 2 years, 4 years, 6 years, every even-numbered age beyond
zero (newborns don’t give birth that year). In the style of Fibonacci,
one could consider each org the equivalent of a breeding pair of
genetically engineered rabbits. But I’ll not mention rabbits or sexual
reproduction again here.

We start with one org, named Eve, late in year zero.  So at the end of
year zero, there is one new org (created, all subsequently are born),
and one org total.

At the end of year 1, Eve is 1 year old, and highly fertile, so she
gives birth to 3 identical triplets, call them Evea, Eveb, and Evec.
So in year 1, there are 3 new orgs, and one old org (Eve) for a total
of 4.

At the end of year 2, Eve is only slightly fertile, and gives birth to
one new clone, called Eved, the only singlet-of-singlet. The triplets
Evea, Eveb, and Evec are each 2 year old, and slightly fertile, so
each gives birth to one child, call them Eveaa, Eveba, and Eveca. So
we have Eve from year 0, triplets from year 1, and 4 new orgs, for a
total at end of year 2: 1 + 3 + 4 = 8.

At the end of year 3, Eve is an odd 3 years old, and highly fertile
again, so she gives birth to 3 identical triplets, call them Eve3a,
Eve3b, and Eve3c. The year-1 triplets Evea, Eveb, and Evec have each
given birth to singlets, and now those singlets (Eveaa, Eveba, and
Eveca) are each an odd 1 year old, and highly fertile.  So each gives
birth to a litter of triplets. Call Eveaa’s triplets Evaaa, Evaab, and
Evaac.  Call Eveab’s triplets Eveaba, Eveabb, and Eveabc, and so
forth. So there are now 24 newborns: one set of triplets of a singlet
child of Eve; three sets of triplets of singlets of triplets of Eve;
and three sets of triplets of triplets of Eve. So there are now a
total of the old 8 plus the new 24 for 8+24 = 32 orgs in the Garden of
Orgeden.

At the end of year 4, Eve is only slightly fertile, being even-aged,
and gives birth to one new clone. Eve’s triplets born in year 1 are
all very fertile odd-aged 3-year-olds, and thus produce 9 offspring.
The three singlet offspring from year 2 of the first-ever triplets,
are now all very fertile odd-aged 1-year-olds, and thus produce 3x3=9
offspring.
The three sets of triplets of triplets born in year 3 are only
slightly fertile, being even-aged (2) and so yield 9 more orgs,
singlets of triplets of triplets.
So the total org population at end of year 4 is one 4-year-old (Eve),
three 3-year-olds (the eldest triplets), four 2-year-olds (one singlet
of Eve and the three singlets of triplets of singlets), 24
1-year-olds, and 29 newborns (Eve’s latest singlet, Eve’s latest
singlet, plus three singlets of triplets of Eve born in Year 1, plus
three singlets of triplets of the singlet of Eve born in Year 2, plus
a singlet of that singlet of Eve born in Year 2, plus three singlets
of the triplets of Eve born in Year 3 for a total of:
1 + 3 + 4 + 24 + 29 = 61 orgs alive.

[tables and unchecked versions of recurrence omitted here]

Origin: this calculation was prompted by a poorly worded summary of
two papers or chapters which I have not yet read [library
unavailable].

http://plato.stanford.edu/entries/fitness/

The summary is:
5. Biological Problems of the Propensity Interpretation
… “there are many circumstances in which the organism of greater
fitness has the propensity to leave fewer immediate offspring than the
organism of lower fitness; as when for example, the larger number of a
bird's chick all die owing to the equal division of a quantity of food
which would have kept a smaller number viable. More generally, as
Gillespie (1977) has shown, the temporal and/or spatial variance in
number of offspring may also have an important selective effect. To
take a simple example from Brandon (1990). If organism a has 2
offspring each year, and organism b has 1 offspring in odd numbered
years and 3 in even numbered ones, then, ceteris paribus, after ten
generations there will be 512 descendants of a and 243 descendants of
b. The same holds if a and b are populations, and b's offspring vary
between 1 and 3 depending on location instead of period.”

Gillespie, G.H., 1977, “Natural selection for variances in offspring
numbers: a new evolutionary principle,” American Naturalist, 111:
1010–1014.

Brandon, R., 1990, Adaptation and Environment. Princeton: Princeton
University Press.

Ekbohm, G., Fagerstrom, T., and Agren, G., 1980, “Natural selection
for variation in offspring numbers: comments on a paper by J.H.
Gillespie,” American Naturalist, 115: 445–447.

In working to reconstruct Gillespie and Brandon from the summary, I deduce:
* they start counting with year 1 (rather than as I do at year 0);
* A clearer rewording would be: “If organism a has 2 offspring each
year, and organism b has 1 offspring in odd numbered years and 3 in
even numbered ones, then, ceteris paribus, AT THE END OF EXACTLY ten
generations there will be 512 descendants of a and 243 newborn
descendants of b. The total If organism a has 2 offspring each year,
and organism b has 1 offspring in odd numbered years and 3 in even
numbered ones, then, ceteris paribus, after ten generations there will
be 512 descendants of a and 243 descendants of b. numbering from year
zero, is as shown below.”

n	Total population at the end of n years
0	0+1 = 1
1	3+1 = 4
2	4+3 = 7
3	7+9 = 16
4	16+9 = 25
5	25+27 = 52
6	52+27 = 79
7	79+81 = 160
8	160+81 = 241
9	241+243 = 484
10	484+243 = 727

This is a sequence in OEIS, A162436, whose partial sum is also in OEIS
as A164123.

I’ll note that the subsequence of primes in the partial sum begins: 7,
79, 241, 727, 19681, …

My new sequences, I think more interesting, have the fertility of
organisms oscillate annually from 1 to 3 or vice versa, but based on
the calendar AGE of the organism, rather than the calendar year
itself.

The two new seqs are:
* The number of newborns in year n under the age-based fertility rule
* The partial sum of the above, namely the Total population at the end
of n years

There are various derived sequences, but I’ll leave those for another time.

-- Jonathan Vos Post