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Sequence MCS01: Rilybeast Breeding Numbers    

Sequence MCS8041809 (formerly MCS01, see the MCS page) begins: 3, 7, 27, 143, 1011, 9111, 100235, 1303071, 19546083, 332283431, 6313385211, 132581089455, ...

The numbers 3, 7, 27 and 143 have long been my favorites, for many reasons not the least of which is related to rilybeasts.

Rilybeasts are from Yom (the second planet of π Ursa Majoris), first explored in 2310 by the crew of the Boreal. They and the early colonists paid most of their attention to triangulus dexteralus (the trups) because of their larger size and considered them the only intelligent native species on the planet (for more details see Handbook for Space Pioneers [3]). It was not until about 80 years later, when the behavioral psychology unit at University Praetolia, on field work in Phecda observed the complex herding patterns described here. It was soon recognized that the rilius species have long surpassed triangulus dexteralus in most measures of intelligence, and that the common rilybeasts (rilius familiaris) posess a far more developed culture. Once this was understood, further human development on Phecda was halted and the current noninterference policies instituted.

In the wild, familiaris is found most often in groups of 3, 7 or 27. On occasion larger groups have been seen, which on close inspection have either been found to number exactly 143, or to be in the thousands.

It is well known that rilybeasts spontaneously appear in groups of three when needed1. This spontaneous appearance in trios is seen in the domesticated rilybeast (rilius domesticus) as well as in each of the wild species. Three is the best number for breeding, as rilybeasts have three-gamete reproduction and triple chromosomes, although all three genders need not be present for fertilization to occur. The trio is by far the most commonly-seen grouping in the wild.    * * Y Y Y Y Y Y Y Y Y Y * Y Y Y Y Y Y * Y Figure 1: The trio just before (left), during, and after trifission into a pack. All are oriented as if traveling towards the top of the page.

After an interval commonly (though inaccurately) called the gestation period (which is usually 27 Yar days long) there will be 7 rilybeasts. Although they are more often found in trios, 7 is the next most often seen number and is called a pack. When traveling in packs the rilybeasts will usually make the formation shown at the right in figure 1. The central creature commands the others; the three immediately adjacent to it focus on watching (for danger or for a sought objective); and the three outermost creatures execute most of the attack or defence when needed, pick out the route on difficult terrain, etc. Due to these roles the rilybeasts' words for heart/mind, eyes and hands/claws have dual meaning, referring also to the three roles within the pack.    Y Y Y Y Y * * * C a b c Y Y Y * Y * * * Y Y Y Y Y C B a b c Y *Y* Y Y * Y Y Y Y * Y Y Y Y Y C B A a b c Y * * Y * * Y Y Y * B A 1 1 1 Y Y Y * * Y Y * A 2 2 2 Y Y * Y Y * 3 3 3 Figure 2:The pack (left) reorganizes itself to prepare for trifission into a tribe. High-speed photography has shown that the growth occurs in three stages as shown here — first the central creature becomes three; then three rows of three appear parallel to the older ones, and finally another three rows of three appear. Thus the number grows from 7, to 9, to 18, to 27; the entire process takes less than a second. The rightmost figure shows how the group divides itself into three nonets (ABC, abc and 123) and thence into nine trios. (In about half of the cases the formation is a mirror-image of what is shown here.)

At the end of the second breeding interval, their number grows to 27, which of course is 3×3×3. This group, called a tribe remains together for another breeding interval. When traveling or performing tasks they organize themselves in trios, or in trios of trios called nonets. At the end of the tribe's breeding interval, disbanding occurs. Disbanding usually consists of individual trios spontaneously popping out of existence, the reverse of the process by which the initial trio appeared. The group will reduce its number to 3, or 6 or 9 or however many are needed given available territory. The resulting trios invariably separate from the others, and result in separate packs/tribes on subsequent generations.

In rare cases the entire tribe (all nine trios) will remain extant, in which case it stays in a unified group, and multiplies into an army of 143 creatures. The army's organization is large and complex. Field researchers report seeing them in platoons of 13, in arrangements resembling the pack and nonet (see figure 3, below). After another breeding cycle the army grows in number to over a thousand, and on the few cases when this has been seen, they organize as trios and occasionally nonets. The group disbands over the course of the next cycle or two, with individual trios disappearing or breaking off independently.

It has recently been discovered, through careful analysis of gamete and fertility cycles, that these numbers 3, 7, 27 and 143 follow a formula that remains invariant throughout the many breeding cycles of a rilybeast's lifetime. The formula can be described mathematically as follows:

X, the total number of creatures, starts at -1.
N, the cycle number, starts at 0 and increases by 1 each breeding cycle.
On each cycle, X is replaced by (X+1)(2N-1)+3.

A bit of explanation is required for the first term: whereas positive numbers represent actual rilybeasts, the negative number -1 represents the need for a rilybeast. 0 would represent nothing at all, neither existence nor need. Thus, the first term must be negative because rilybeasts pop into existence only when needed.

If you calculate this formula for several cycles, you get the values:

N=0, X=-1
N=1, X=3
N=2, X=7
N=3, X=27
N=4, X=143
N=5, X=1011
N=6, X=9111

and so on. Since the discovery of the formula there has been much speculation about its origin, purpose and implications, across multiple disciplines.

Those who study the group psychology and culture of the rilybeasts note the symbolic significance in the numbers. The first number, 3 is so important it is almost trivial to mention; it is almost too common to have spiritual meaning. The number 7 represents consciousness and the triumph of intelligence; it is the sum of the heart/mind, eyes and hands/claws (1+3+3), which are also the roles within the pack (as described above). Almost all rilybeasts share the memory of being part of a pack of 7. The importance of 27 is obvious, being 3×3×3, three threes multiplied together. 143 is a symbol of society (a large number of individuals, with a social structure more complex than the trio, pack and nonet) — because it cannot be divided by 3, 7 or 9: 143=11×13. These factors have additional significance because they are three 1's and one 3. This is interpreted in different ways, such as the trio viewed as individuals followed by the trio as a unit, or the three elements of life (feeling, awareness, action) coming together. The next number in the sequence, 1011, is three 1's, and the 1's appear in the 1st, 4th and 3rd digits. Furthermore, when you factor this number you get 3×337: three 3's and a 7. The next number in the sequence is 9111, a 9 and three 1's. 9 of course is the nonet, 3+3+3, the next most common grouping in the rilybeast collective consciousness after the trio and pack. 9111 when factored gives 3×3037, which is again three 3's and a 7. You can also add the digits to get significant numbers: 2+7 gives 9; 1+0+1+1 gives 3, 9+1+1+1 gives 12 and 1+2 gives 3. The same process with the factors 3×337 (or 3×3037) yields 7. All of these facts hold symbolic and spiritual significance to the rilybeasts.

Genetic researchers look strictly at origin: how did the formula come to be? They have suggested (based on TNA evidence from other species that share a common ancestor with rilius) that this formula results from three successive and distinct genetic developments. All early species had the ability to produce one offspring per cycle, following the formula X'=X+1. Later, when evolution brought sexual reproduction, three young were produced at at a time, and the formula became X'=X+3. Then, so the theory says, shortly before vertebrate diversity began, new mutations produced the ability to breed by a rate that increased over time. At whatever rate the first generation produced (call this "1x"), the rate grew to 3 times as great during the second generation, then to 5 times the original rate during the third generation, and so on for however long as a breeding trio was viable. This increase from 1x to 3x to 5x was represented by the expression 2N-1, where N is the generation number. The mechanism for this is the subject of intense debate — it results in such explosive population growth that it can only have occurred concurrently with (or after) the development of spontaneous trifusion, the phenomenon by which rilybeasts suddenly disappear in groups of three at a time — but this is thought to have evolved significantly later. Some theories propose that the lifetime, or at least the breeding (fertile) portion of the lifetime was much shorter. Regardless, it is agreed that all three breeding formulas were present in the genes of the advanced threemals, and became activated concurrently in Trimagnon rilius. Thus the current formula, X'=(X+1)(2N-1)+3, which clearly incorporates the early X+1, times the rate multiplier 2N-1, and the +3 from the intermediate formula.

Zoologists look at the physical implications of the formula, in particular, how it benefits the survival of the species. To gain insight on this, they mainly point to the prime factorizations of the numbers, already mentioned above, because they determine how a group can break up into symmetrical smaller groups for organizational purposes. The obvious strengths of the trio grouping as seen in the wild show how important it is for the formula to generate numbers that are multiples of 3. Any numbers that are not a multiple of three should at least be divisible into numbers that can be arranged in three-way symmetry, because of their obvious organizational benefit. The pack and platoon are examples of this. There is also speculation that it is useful that the larger numbers (27, 143, 1011 and 9111) factor in only one way, rather than in multiple ways like "human" grouping numbers e.g. 24, 144, 1000 etc.    y y Y Y Y Y Y Y y Y Y y Y Y Y Y Y Y Y Y Y Y y y Y Y Y Y Y Y y Y Y Y Y Y Y y y trio packs nonet platoon (standard,alternate) (part of tribe, (part of army, (generation 2) generation 3) generation 4) Figure 3: The rilybeast grouping units are shown, in order of importance. The progression from left to right also follows the frequency of occurrence (most frequent on the left) and the chronological sequence of a typical individual's life (earliest on the left). The platoon of 13 pictured at right is particularly complex. Each large 'Y' shows where the more experienced individuals will position themselves. It functions both as a pack and as a nonet — the three outermost groups of three behave as three trios, each of which serve the hand/claw pack role, with the one shown as "Y" directing the other two (as they would in a nonet); the remaining 4 serve purely in the heart/mind and eye roles of a pack.

Factorizations of the first 27 terms in the sequence:
3 = 3
7 = 7
27 = 33
143 = 11 × 13
1011 = 3 × 337
9111 = 3 × 3037
100235 = 5 × 20047
1303071 = 3 × 7 × 11 × 5641
19546083 = 33 × 389 × 1861
332283431 = 332283431
6313385211 = 3 × 7 × 59 × 5095549
132581089455 = 3 × 5 × 172 × 30583873
3049365057491 = 1690267 × 1804073
76234126437303 = 3 × 906427 × 28034663
2058321413807211 = 3 × 7 × 41 × 109 × 137 × 160089547
59691321000409151 = 131 × 271 × 1681398298651
1850430951012683715 = 3 × 5 × 13 × 9489389492372737
61064221383418562631 = 32 × 73 × 23 × 167 × 787 × 6543817139
2137247748419649692123 = 11 × 88093 × 32035417 × 68847853
79078166691527038608591 = 3 × 77509 × 187163 × 1817034636691
3084048500969554505735091 = 3 × 1028016166989851501911697
126445988539751734735138775 = 52 × 7 × 29 × 101 × 641 × 1093 × 980831 × 358984139
5437177507209324593610967371 = 3 × 2687 × 58916471063 × 11448480917297
244672987824419606712493531743 = 3 × 113 × 82193 × 40791694787 × 215268133607
11499630427747721515487195991971 = 7 × 2063585154907 × 796092345845131679
563481890959638354258872603606631 = 3 × 23 × 19477 × 20757798811 × 20198890274770717
28737576438941556067202502783938235 = 32 × 5 × 271813436635649 × 2349452689530879367

Some other sequences (all of them much less fictional) are discussed here.


1 : They literally appear out of thin air, as if they have somehow arrived in our plane of existence out of a parallel dimension. See [2], part II, section 15, in which a visiting sphere from a 3rd dimension appears into the narrator's 2-dimensional world: "What was our horror when we saw before us a Figure! [...] it seemed to change its size in a manner impossible for a Circle or for any regular Figure of which I had had experience.".

Some willful suspension of disbelief on the part of readers is needed to imagine Rilybeasts suddenly appearing out of nowhere. It is amusing to note that I, after not having read my own story for a number of years, found it easy to believe the notion of humans colonizing and exploring a distant planet harboring two intelligent alien life-forms, yet balked at the idea of the Rilybeasts appearing suddenly out of thin air.


[2] Edwin A. Abbott, Flatland — a romance of many dimensions. 1884.

[3] Stephen L. Wolfe and Roy L. Wysack, Handbook for Space Pioneers — Guide for Pioneers from Earth to the Eight Planets Now Available for Colonization, Grosset & Dunlap (1978), ISBN 0-448-16178-8.

This book has the original description of the planet Yom and its larger inhabitants.

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