tetrahedral number
Let an agent explain tetrahedral numbers and their formula and properties, connect them to triangular numbers and Pascal triangle, support calculations and problems, and support learning.
Research draft, second pass
A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.
written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify
Researched by: Claude
Purpose and description
Let an agent explain tetrahedral numbers and their formula and properties, connect them to triangular numbers and Pascal triangle, support calculations and problems, and support learning.
A figurate number representing a pyramid with a triangular base and three sides, equal to the sum of the first n triangular numbers and given by the formula n times n plus one times n plus two divided by six, beginning 1, 4, 10, 20, 35, 56 and 84; tetrahedral numbers appear in Pascal triangle, in counting problems such as stacking spheres, and in combinatorics as binomial coefficients.
What it is for: Numbers counting tetrahedral stacks.
It can be explain formula and properties; connect to related numbers; support calculations; support learning.
Distinguishing features
Figurate number
Cubic formula
Pascal triangle column
Sphere stacking
What it looks like
Not physical; a sequence of integers.
How it is recognised
Sum of triangular numbers
Binomial coefficient n plus two choose three
Square pyramidal numbers use square layers
Related models
is a kind of - category
is related to - triangular layers
is related to - another named integer class
is related to - the triangular base of the tetrahedron
In practice
Families and kinds
tetrahedral numbers
triangular numbers as layers
square pyramidal numbers as a related class
pentatope numbers in higher dimension
tetrahedral numbers that are also other figurate numbers
Standards and regulation
Mathematical conventions
No regulation
Failure modes and hazards
Confusing tetrahedral and square pyramidal numbers
Formula errors
Assuming patterns without proof
Where this came from
wikidata · CC0 1.0
Drafted structure
Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.
Understand What tetrahedral numbers are.
Mathematics.
Definition
Definition and formula.
Definition
Definition.
- How are tetrahedral numbers defined as sums of triangular numbers, and how is the formula derived? definition
- Which entry fits triangular numbers? action
Pascal
Pascal triangle and binomials.
Pascal
Pascal.
- How do tetrahedral numbers appear in Pascal triangle and as binomial coefficients? definition
- Which entry fits binomial coefficients? action
Compute Calculations and problems.
Practice.
Calculate
Calculating.
Calculate
Calculation.
- What is the nth tetrahedral number, and is a given number tetrahedral? action
- Which entry fits integer sequences? action
Problems
Counting problems.
Problems
Problems.
- How do tetrahedral numbers solve problems such as stacking cannonballs or counting triples? action
- Which entry fits combinatorics? action
Theory Properties and results.
Number theory.
Properties
Properties.
Properties
Properties.
- What properties and identities do tetrahedral numbers satisfy, and which are also squares or triangular? provenance
- Which questions remain open? boundary
Generalise
Generalisations.
Generalise
Generalisations.
- How do tetrahedral numbers generalise to higher-dimensional simplex numbers? definition
- Which entry fits figurate numbers? action
Learn History and teaching.
Education.
History
History.
History
History.
- How were figurate numbers studied from the Pythagoreans to modern times? provenance
- Which references are standard? provenance
Teach
Teaching.
Teach
Teaching.
- How can tetrahedral numbers be taught with physical stacking? action
- Which misconceptions arise? provenance
What the second pass must settle
- Should figurate numbers be a separate entry?
- How should integer sequence databases be linked?
- How should combinatorics resources be linked?