A pentagonal number counts dots in a nested-pentagon pattern. Starting with a single dot, each next pentagon shares two sides with the previous one and adds a new layer of dots:
P1=1,P2=5,P3=12,P4=22,P5=35,P6=51,…
The closed form is
Pn=2n(3n−1)
Pentagonal numbers are the third in the family of polygonal numbers (after triangular and square). The general r-gonal-number formula
Pn(r)=2n[(r−2)n−(r−4)]
specializes to the pentagonal case at r=5.
Sum-of-arithmetic-series form.Pn=1+4+7+10+⋯+(3n−2), the sum of the first n terms of the arithmetic series with a1=1 and common difference d=3. The first term 1 and the step 3 reflect the geometry: the first pentagon has 1 dot, and each new layer adds 3 dots times n minus an adjustment for the shared vertex.
For deeper coverage see pentagonal number, polygonal numbers, and Euler's pentagonal number theorem.
The Closed Form Derivation
From the arithmetic-series perspective. Pentagonal numbers are sums of the arithmetic series 1,4,7,10,…, where each term is 3k−2 for k=1,2,…,n. The sum is
Pn=k=1∑n(3k−2)=3k=1∑nk−2n=3⋅2n(n+1)−2n
Simplifying:
Pn=23n2+3n−4n=23n2−n=2n(3n−1)
From the general $r$-gonal formula. Substituting r=5 into
Pn(r)=2n[(r−2)n−(r−4)]
gives Pn(5)=n(3n−1)/2.
Why the formula is always an integer. Among any two consecutive integers, one is even. Either n is even, or 3n−1 is even (since 3n has the same parity as n). So n(3n−1) is always even.
The Membership Test
Given a positive integer m, testing whether it is pentagonal requires two checks.
The test.m is pentagonal if and only if both:
1. 24m+1 is a perfect square, and 2. 1+24m+1 is divisible by 6.
If both hold, the index is
n=61+24m+1
Why both checks are needed. From Pn=n(3n−1)/2=m, the quadratic 3n2−n−2m=0 has discriminant 1+24m, which must be a perfect square. But the quadratic formula gives n=(1±24m+1)/6, and we need n to be a positive integer, hence the divisibility condition.
Example. Is 117 pentagonal? Compute 24⋅117+1=2809=532 — a perfect square. Then (1+53)/6=9, an integer. So 117=P9. Verify: 9⋅26/2=117. Confirmed.
Non-example. Is 50 pentagonal? Compute 24⋅50+1=1201. Then 1201≈34.66 — not a perfect square. So 50 is not pentagonal. The nearest are P5=35 and P6=51.
Edge case where check 1 passes but check 2 fails. If 24m+1 happens to be a perfect square but the resulting 1+s is not divisible by 6, the candidate index would be fractional. The divisibility check rejects these.
Properties and Identities
• Recurrence: Pn=Pn−1+(3n−2) with P1=1. Each pentagonal number adds the next arithmetic-series term. • Generating function: ∑n=0∞Pnxn=(1−x)3x(1+x). The pentagonal numbers' generating function is a rational function. • Connection to triangulars: Pn=T3n−1−Tn−1, where Tk is the k-th triangular number. A pentagonal number is a difference of two triangulars. • Connection to hexagonal numbers: every hexagonal number Hn relates to a pentagonal number, though the correspondence is not as clean as Hn=T2n−1 for triangulars. • Asymptotic growth: Pn∼23n2 as n→∞. Pentagonal numbers grow as the square of the index. • Last-digit pattern: pentagonal numbers in base 10 end only in 0, 1, 2, 5, 6, or 7. Numbers ending in 3, 4, 8, or 9 cannot be pentagonal.
Euler's Pentagonal Number Theorem
Pentagonal numbers' most famous mathematical role is in Euler's pentagonal number theorem, a foundational result in the theory of integer partitions.
The theorem. For all formal power series:
n=1∏∞(1−xn)=k=−∞∑∞(−1)kxk(3k−1)/2
The exponents on the right are exactly the generalized pentagonal numbers0,1,2,5,7,12,15,22,26,…, obtained by letting k range over all integers (positive, zero, negative) in the formula k(3k−1)/2.
The combinatorial meaning. The left side is the generating function for the difference between the number of partitions of n into an even number of distinct parts and the number into an odd number of distinct parts. Euler's theorem says this difference is 0 except when n is a generalized pentagonal number, in which case it is ±1.
Why it matters. The reciprocal 1/∏(1−xn) is the generating function for the partition functionp(n). Euler's theorem gives a recurrence:
The signs follow a ++−−++−− pattern, and the subtractions stop when the argument goes negative. This is the fastest classical recurrence for the partition function — computing p(100) requires only about 15 terms.
Common Applications
• Integer partitions. As shown above, pentagonal numbers control the partition function via Euler's recurrence. Any algorithm computing p(n) efficiently goes through pentagonal numbers. • Modular forms. The product ∏(1−xn)=η(τ)/x1/24 (up to a factor) is the Dedekind eta function, a basic modular form. Pentagonal numbers appear in its q-expansion. • Combinatorial identities. Many partition identities, including Jacobi's triple product, build on Euler's pentagonal-number theorem. • Geometric counting. Pentagonal numbers count dots in nested-pentagon patterns; the dual of triangular and square configurations. • Lattice problems. Some 2D lattice-counting problems on pentagonal regions or 5-fold-symmetric structures involve pentagonal-number boundaries. • Recreational mathematics. Pentagonal numbers appear in puzzles, sequences, and figurate-number problems where one extends the polygon-counting paradigm beyond triangles and squares.
Common Mistakes
• Confusing pentagonal with pentagon-related counts. Pentagonal numbers count dots in a specific nested-pentagon pattern, not all pentagonal arrangements. The number of dots in a regular pentagon's lattice differs depending on the construction.
• Forgetting the divisibility check.24m+1 being a perfect square is necessary but not sufficient. The divisibility-by-6 condition on 1+24m+1 filters out spurious candidates. Example: if 24m+1=49=72, then 1+7=8, not divisible by 6 — so m=2 is not pentagonal even though the perfect-square check passes.
• Confusing with generalized pentagonal numbers. The "classical" pentagonal numbers use n≥1 giving 1, 5, 12, 22, ... The generalized version allows all integer n giving 0, 1, 2, 5, 7, 12, 15, ... — different sequences with different roles.
• Misapplying the closed form.Pn=n(3n−1)/2 is exact for positive integer n. Extending to fractional or negative n requires the generalized formulation.
• Confusing $P_n$ with the partition function $p(n)$. Both use the letter p, but p(n) counts integer partitions while Pn is the n-th pentagonal number. The two are connected via Euler's theorem but distinct.
Related Sequences and Concepts
Triangular Numbers — Tn=n(n+1)/2. The first figurate family; pentagonal numbers extend the pattern to 5 sides.
Square Numbers — Sn=n2. The second figurate family; sits between triangular and pentagonal.
Hexagonal Numbers — Hn=n(2n−1). The next polygonal sequence after pentagonal.
Generalized Pentagonal Numbers — 0, 1, 2, 5, 7, 12, 15, 22, ... — pentagonal-number formula extended to all integer indexes. Appears in Euler's theorem.
Integer Partition — the number of ways to write n as a sum of positive integers, disregarding order. Pentagonal numbers determine the recurrence for the partition function.
Partition Function $p(n)$ — counts partitions of n. Euler gave a pentagonal-number recurrence; Hardy and Ramanujan gave an asymptotic; Rademacher gave an exact formula.
Dedekind Eta Function — a modular form whose q-expansion features pentagonal-number exponents. Foundational in the theory of modular forms.
Polygonal Numbers — the family of r-gonal numbers for r=3,4,5,6,…. Pentagonal is the r=5 case.
Euler's Pentagonal Number Theorem — the identity ∏(1−xn)=∑(−1)kxk(3k−1)/2, summed over all integers k. One of the most beautiful results in the theory of partitions.
Figurate Numbers — the umbrella term covering all polygonal, pyramidal, and higher-dimensional analogs. Pentagonal is one entry in a vast taxonomy.
Generating Function — the formal power-series tool that connects pentagonal numbers to partitions, modular forms, and combinatorial identities.