Somos sequence¶

In mathematics, a Somos sequence is a sequence of numbers defined by a certain recurrence relation, described below.

Version: v1 · 2026-09-28 · History
Domain-specific #: 12165
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Number Theory, Recurrence Relations → Mathematics

Core Idea¶

Somos sequence is treated here as the recurring mathematics, logic, and statistics identity summarized by this source-grounded definition: In mathematics, a Somos sequence is a sequence of numbers defined by a certain recurrence relation, described below.

In mathematics, a Somos sequence is a sequence of numbers defined by a certain recurrence relation, described below. They were discovered by mathematician Michael Somos. From the form of their defining recurrence (which involves division), one would expect the terms of the sequence to be fractions, but surprisingly, a few Somos sequences have the property that all of their members are integers.

For k = 2 or 3 , the defining relations are very simple (there is no addition on the right-hand side). The (x, y) coordinates of the sequence Q + n P are given by (\wp(z_0 + n \kappa), \wp'(z_0 + n \kappa)) , where \wp(z) = \wp(z; g_2, g_3) denotes the Weierstrass elliptic function. For an integer number k larger than 1 , a Somos- k sequence {a_n}_{n \in \mathbb{Z}} is a solution of the equation.

For Somos sequence, the abstraction is narrower than the article's general subject matter: a positive case must preserve In mathematics, a Somos sequence is a sequence of numbers defined by a certain recurrence relation, described below. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in mathematics, logic, and statistics, which is why this identity is domain-specific rather than prime.

Structural Signature¶

Sig role-phrases:

Defining carrier — Hence a non-degenerate solution is determined by a choice of k initial values a_0 , a_1 , ..., a_{k - 1} .
Constitutive relation — The sequence yielded by setting \alpha_1 = \alpha_2 = \dots = \alpha_{\lfloor k/2 \rfloor} = 1 and a_0 = a_1 = \dots = a_{k - 1} = 1 is referred to as the Somos- k sequence.
Operating condition — The form of the recurrences describing the Somos sequences involves divisions, making it appear likely that the sequences defined by these recurrence will contain fractional values.
Recognition evidence — The six parameters g_2, g_3, A, B, z_0, \kappa \in \mathbb{C} are determined uniquely (up to a choice of sign) by the coefficients \alpha, \beta and the initial terms a_0, a_1, a_2, a_3 .
Admissible variation — The (x, y) coordinates of the sequence Q + n P are given by (\wp(z_0 + n \kappa), \wp'(z_0 + n \kappa)) , where \wp(z) = \wp(z; g_2, g_3) denotes the Weierstrass elliptic function.
Characteristic consequence — In mathematics, a Somos sequence is a sequence of numbers defined by a certain recurrence relation, described below.
Failure boundary — They were discovered by mathematician Michael Somos.

What It Is Not¶

Not the whole field of mathematics, logic, and statistics. The node requires the specific identity stated by In mathematics, a Somos sequence is a sequence of numbers defined by a certain recurrence relation, described below.
Not an over-broad reading. For an integer number k larger than 1 , a Somos- k sequence {a_n}_{n \in \mathbb{Z}} is a solution of the equation.
Not an over-broad reading. a_n a_{n-k} = \sum_{i=1}^{\lfloor k/2 \rfloor} \alpha_i a_{n-i} a_{n-k+i}.
Not an over-broad reading. It can be rearranged into the form of an order k recurrence relation.
Not automatically Elliptic Divisibility Sequence. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Somos sequence applies literally inside mathematics, logic, and statistics wherever the source-defined carrier and relation can be established. Its documented habitats include:

Integrality and the Laurent phenomenon. That is, as a function of the initial terms a_0 , ..., a_{k-1} , every term a_n is a multivariate Laurent polynomial with coefficients in \mathbb{Z}[\alpha_1, \dots, \alpha_{\lfloor k/2 \rfloor}] .
The general term is given by the formula. where \sigma(z) = \sigma(z; g_2, g_3) denotes the Weierstrass sigma function associated with the curve E written in the canonical form.
The general term is given by the formula. The (x, y) coordinates of the sequence Q + n P are given by (\wp(z_0 + n \kappa), \wp'(z_0 + n \kappa)) , where \wp(z) = \wp(z; g_2, g_3) denotes the Weierstrass elliptic function.
The general term is given by the formula. The coefficients \alpha and \beta are given as elliptic functions of \kappa by.
Recurrence equations. For an integer number k larger than 1 , a Somos- k sequence {a_n}_{n \in \mathbb{Z}} is a solution of the equation.
Recurrence equations. a_n a_{n-k} = \sum_{i=1}^{\lfloor k/2 \rfloor} \alpha_i a_{n-i} a_{n-k+i}.

Outside mathematics, logic, and statistics, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Classification or should be marked as analogy.

Clarity¶

A clear use of Somos sequence names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In mathematics, a Somos sequence is a sequence of numbers defined by a certain recurrence relation, described below. The strongest recognition evidence in the frozen account is: The six parameters g_2, g_3, A, B, z_0, \kappa \in \mathbb{C} are determined uniquely (up to a choice of sign) by the coefficients \alpha, \beta and the initial terms a_0, a_1, a_2, a_3 . A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification For an integer number k larger than 1 , a Somos- k sequence {a_n}_{n \in \mathbb{Z}} is a solution of the equation. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Somos sequence compresses multiple mathematics, logic, and statistics details into a stable diagnostic relation. The source shows both the central mechanism—the sequence yielded by setting \alpha_1 = \alpha_2 = \dots = \alpha_{\lfloor k/2 \rfloor} = 1 and a_0 = a_1 = \dots = a_{k - 1} = 1 is referred to as the Somos- k sequence.—and the practical consequence—in mathematics, a Somos sequence is a sequence of numbers defined by a certain recurrence relation, described below. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the mathematics, logic, and statistics entities to which the claim applies.
State the relation. Use the source-grounded identity: In mathematics, a Somos sequence is a sequence of numbers defined by a certain recurrence relation, described below.
Check operation and conditions. The form of the recurrences describing the Somos sequences involves divisions, making it appear likely that the sequences defined by these recurrence will contain fractional values.
Demand recognition evidence. The six parameters g_2, g_3, A, B, z_0, \kappa \in \mathbb{C} are determined uniquely (up to a choice of sign) by the coefficients \alpha, \beta and the initial terms a_0, a_1, a_2, a_3 .
Test variation. Change an implementation or setting while preserving the (x, y) coordinates of the sequence Q + n P are given by (\wp(z_0 + n \kappa), \wp'(z_0 + n \kappa)) , where \wp(z) = \wp(z; g_2, g_3) denotes the Weierstrass elliptic function.
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Classification.

Knowledge Transfer¶

Within the home domain. Knowledge about Somos sequence transfers literally when a new case preserves the same carrier type, relation, and recognition test. That is, as a function of the initial terms a_0 , ..., a_{k-1} , every term a_n is a multivariate Laurent polynomial with coefficients in \mathbb{Z}[\alpha_1, \dots, \alpha_{\lfloor k/2 \rfloor}] . where \sigma(z) = \sigma(z; g_2, g_3) denotes the Weierstrass sigma function associated with the curve E written in the canonical form.

Beyond the home domain. No canonical parent is asserted for Somos sequence. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples¶

Canonical¶

In the first nontrivial case, k = 4 , the relation is. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → In mathematics, a Somos sequence is a sequence of numbers defined by a certain recurrence relation, described below; recognition evidence → The six parameters g_2, g_3, A, B, z_0, \kappa \in \mathbb{C} are determined uniquely (up to a choice of sign) by the coefficients \alpha, \beta and the initial terms a_0, a_1, a_2, a_3

Applied / In Practice¶

In the case k = 5 the relation is. The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → Recurrence equations; invariant → In mathematics, a Somos sequence is a sequence of numbers defined by a certain recurrence relation, described below; boundary → the case exits the class when for an integer number k larger than 1 , a Somos- k sequence {a_n}_{n \in \mathbb{Z}} is a solution of the equation

Structural Tensions¶

T1 — Stable identity versus admissible variation. For an integer number k larger than 1 , a Somos- k sequence {a_n}_{n \in \mathbb{Z}} is a solution of the equation. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. a_n a_{n-k} = \sum_{i=1}^{\lfloor k/2 \rfloor} \alpha_i a_{n-i} a_{n-k+i}. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. It can be rearranged into the form of an order k recurrence relation. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. a_n = \frac{\sum_{i=1}^{\lfloor k/2 \rfloor} \alpha_i a_{n-i} a_{n-k+i}}{a_{n-k}}. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. Hence a non-degenerate solution is determined by a choice of k initial values a_0 , a_1 , ..., a_{k - 1} . The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Somos sequence literally, co-instantiate Classification, or only resemble it?

T6 — Autonomy versus reduction. The sequence yielded by setting \alpha_1 = \alpha_2 = \dots = \alpha_{\lfloor k/2 \rfloor} = 1 and a_0 = a_1 = \dots = a_{k - 1} = 1 is referred to as the Somos- k sequence. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Somos sequence distinguish that the broader parent Classification leaves together?

Structural–Framed Character¶

Somos sequence is structural-leaning. Its structural side is the repeatable organization summarized by In mathematics, a Somos sequence is a sequence of numbers defined by a certain recurrence relation, described below. Its framed side is the mathematics, logic, and statistics vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: The form of the recurrences describing the Somos sequences involves divisions, making it appear likely that the sequences defined by these recurrence will contain fractional values. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Classification. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. In mathematics, a Somos sequence is a sequence of numbers defined by a certain recurrence relation, described below. The stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: Hence a non-degenerate solution is determined by a choice of k initial values a0 , a1 , ..., a{k - 1} . The sequence yielded by setting \alpha1 = \alpha2 = \dots = \alpha{\lfloor k/2 \rfloor} = 1 and a0 = a1 = \dots = a{k - 1} = 1 is referred to as the Somos- k sequence. It further constrains recognition and variation through: The form of the recurrences describing the Somos sequences involves divisions, making it appear likely that the sequences defined by these recurrence will contain fractional values. The six parameters g2, g3, A, B, z0, \kappa \in \mathbb{C} are determined uniquely (up to a choice of sign) by the coefficients \alpha, \beta and the initial terms a0, a1, a2, a3 .

What is domain-bound. mathematics, logic, and statistics supplies the operative entities, technical vocabulary, warrants, and exceptions that make Somos sequence literal. Its documented scope includes the condition that That is, as a function of the initial terms a0 , ..., a{k-1} , every term an is a multivariate Laurent polynomial with coefficients in \mathbb{Z}[\alpha1, \dots, \alpha{\lfloor k/2 \rfloor}] . Another bounded application condition is that where \sigma(z) = \sigma(z; g2, g3) denotes the Weierstrass sigma function associated with the curve E written in the canonical form. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—The (x, y) coordinates of the sequence Q + n P are given by (\wp(z0 + n \kappa), \wp'(z0 + n \kappa)) , where \wp(z) = \wp(z; g2, g3) denotes the Weierstrass elliptic function.—and future graph densification may discover a defensible relation only if it preserves that boundary.

This entry presupposes Recurrence relation.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Somos sequence. The reviewed identity is: In mathematics, a Somos sequence is a sequence of numbers defined by a certain recurrence relation, described below. The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

Relationships to Other Abstractions¶

Current abstraction Somos sequence Domain-specific

Parents (1) — more general patterns this builds on

Somos sequence presupposes Recurrence relation Domain-specific

A Somos sequence is generated by a defining recurrence relation.

Hierarchy path (1) — routes to 1 parentless root

Somos sequence → Recurrence relation → Recursion

Neighborhood in Abstraction Space¶

Somos sequence sits in a moderately populated region (45^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Unclustered & Miscellaneous (2551 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Classification. The parent omits the specialist differentia. Tell: Can the case establish In mathematics, a Somos sequence is a sequence of numbers defined by a certain recurrence relation, described below?
Elliptic Divisibility Sequence. An integer divisibility sequence generated by the nonlinear recurrence of elliptic-curve division polynomials, translating multiplication of a rational point into term divisibility, height growth, ranks of apparition, and primitive-divisor structure. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Sequence number. A sequence number is an identifier assigned according to an ordering rule so receivers can distinguish instances, reconstruct order, detect gaps or duplicates, and coordinate state across a stream or collection. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Golomb sequence. The nondecreasing self-describing integer sequence in which each positive integer n occurs exactly as many times as the nth term states. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Somos sequence remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside mathematics, logic, and statistics lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Classification?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Somos_sequence (revision 1361565842).
Preserved source candidate: https://www.kurims.kyoto-u.ac.jp/~kenkyubu/bessatsu/open/B41/pdf/B41_003.pdf
Preserved source candidate: https://faculty.uml.edu/jpropp/somos/chronology.html
Preserved source candidate: https://www.quantamagazine.org/the-astonishing-behavior-of-recursive-sequences-20231116/
Preserved source candidate: http://faculty.uml.edu/jpropp/somos.html
Preserved source candidate: https://www.numberphile.com/videos/the-troublemaker-number

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.