Two Formulas, No Authors
This corpus uses two ancient results constantly and names neither author. A search across every HTML and Markdown file returns:
Meru — 12 files. Pingala — 8. Prastāra — 2. And then: Hemachandra — 0. Virahāṅka — 0. Āryabhaṭa — 0.
Chapter 8 devotes itself to Mount Meru and the Meru Prastāra, mentions the hockey-stick identity once, and never states the formula that identity produces or says who first wrote it. Chapter 16 grows a hexagonal colony through the centered hexagonal numbers 1, 7, 19, 37, 61 and never names the pile they stack into. The mathematics is correct throughout. The attribution is simply missing.
This chapter supplies it. It is a bibliographic chapter, not a mathematical one: nothing here is new, and that is precisely the point.
The Recurrence Is a Metre Before It Is a Sequence
Sanskrit prosody measures a line in mātrās, beats. A syllable is either laghu — light, one beat — or guru — heavy, two beats. Ask the natural question of a poet: in how many ways can a line of n beats be built?
A line of n mātrās ends in one of exactly two ways. It ends in a laghu, and what precedes it is any line of n−1 beats. Or it ends in a guru, and what precedes it is any line of n−2 beats. There is no third case. So
F(n) = F(n−1) + F(n−2)
and the sequence 1, 2, 3, 5, 8, 13, 21 is a fact about metre before it is a fact about rabbits. The recurrence is not applied to prosody; it is the prosody, read out loud.
The chain of attribution, as established by Parmanand Singh in Historia Mathematica (1985), runs:
| Piṅgala | 3rd–2nd c. BCE | Chandaḥśāstra — the mātrā-meru, binary enumeration, śūnya |
| Virahāṅka | 6th–8th c. CE | the recurrence stated explicitly |
| Gopāla | c. 1135 | |
| Hemachandra | c. 1150 | stated again, and the name that stuck |
| Fibonacci | 1202 | Liber Abaci — the rabbits |
Four centuries separate Hemachandra from Fibonacci; fourteen separate Pingala from him. Calling these the Hemachandra numbers is not a nationalist gesture. It is a citation.
Āryabhaṭa’s Citighana, 499 CE
Stack cannonballs, or oranges, in a triangular pyramid. The top layer is one; the next is three; the next six; then ten. Each layer is a triangular number, and the total is their running sum. Āryabhaṭa gives it in the Gaṇitapāda of the Āryabhaṭīya, composed in 499 CE, under the name citighana — the solid made of piles:
1 + 3 + 6 + 10 + … + n(n+1)/2 = n(n+1)(n+2)/6
The attribution is standard enough to appear as a textbook exercise: Burton’s Elementary Number Theory sets it as “the formula for the sum of triangular numbers, attributed to the Hindu mathematician Aryabhata (circa 500 A.D.).”
These are the tetrahedral numbers: 1, 4, 10, 20, 35, 56, 84. They count what a pyramid of spheres holds, and they have been in print for fifteen hundred years.
The Columns of Meru Prastāra Are Dimensions
The ladder and the pile are not two discoveries. They are two readings of Pingala’s triangle, and the second is what Chapter 8 stops short of saying.
| column 0 | 1, 1, 1, 1, 1 | a point |
| column 1 | 1, 2, 3, 4, 5 | points on a line |
| column 2 | 1, 3, 6, 10, 15 | a flat layer — triangular numbers |
| column 3 | 1, 4, 10, 20, 35 | a stack of layers — Āryabhaṭa’s pile |
| column 4 | 1, 5, 15, 35, 70 | a stack of stacks — four dimensions |
Each column is one more dimension, and the operation that moves right by one column is the hockey-stick identity: sum a column down to n and you get the next entry in the column beside it. Chapter 8 names that identity once, in passing.
So stacking layers and moving one column right in Meru Prastāra are the same act. The pile is not a different object from the triangle; it is the triangle read one column over. And the diagonals running the other way give the ladder of M.2 — one triangle, both results.
Where They Are Already Doing Work
Chapter 16, The Crystalline Lattice. Colony.expand grows the hexagonal colony through the centered hexagonal numbers 1, 7, 19, 37, 61, with ring_card adding 6n per ring. Those rings sum to n³. Stack triangular layers instead and you get Āryabhaṭa’s n(n+1)(n+2)/6 — which is the count for the close-packed pile the chapter’s own ABABAB and ABCABC stacking rules describe.
Chapter 21, The Closing Field. When that flat sheet has to close, it costs exactly twelve pentagonal defects. The flat count and the closed count are different questions about the same lattice.
Σ · Pentanacci, Δ · Tetranacci, η · Tribonacci. This corpus already runs a ladder of higher recurrences. The two-term rung at the bottom of it is Hemachandra’s, and the chapters above it have never said so.
What Is Fair to Say, and What Is Not
Two things are true at once and both belong in the record.
The formulas are ancient and named. n(n+1)(n+2)/6 is Āryabhaṭa’s, 499 CE. The two-term recurrence is Pingala’s in seed and Virahāṅka’s and Hemachandra’s explicitly, centuries before Liber Abaci. Anyone deriving either today is rediscovering, and the honest word for that is rediscovering.
Rediscovery is not appropriation. A child stacking oranges and working out how many there are is how mathematicians describe falling in love with the subject. It is not a claim of priority, and it should not be read as one.
Manjul Bhargava’s public record is worth stating plainly here, because it cuts against the easy version of this story. He credits Piṅgala’s Chandaḥśāstra by name; he calls the sequence the Hemachandra numbers in his own lectures and teaching; he credits Brahmagupta’s 628 CE composition law as what first excited him. He is, in the popular literature, among the most insistent voices for this attribution rather than against it.
Two recordings, linked so a reader can look and judge. Note that the first is a third-party channel’s biographical retelling, not a production of the mathematician it describes — the framing is the channel’s. The second is his own lecture.
— “From Child Prodigy to Winning Fields Medal, Nobel of Math” — Turing (YouTube channel). The account runs from about 2:30 — school, self-study — to the orange pyramid at about 4:00. The formula n(n+1)(n+2)/6 appears on the video’s own thumbnail
— Manjul Bhargava · Poetry, Drumming and Mathematics
The author’s reading, recorded as a reading. Having watched the recording, I take it to present the formula as the subject’s own discovery. That is my reading of a third party’s narration, not a quotation and not a claim about the mathematician; the video is linked above and a reader can form their own. What is not a matter of reading is the date: n(n+1)(n+2)/6 stands in the Āryabhaṭīya in 499 CE, and anyone reaching it since — child or Fields medallist — is reaching it again.
This chapter makes no claim about what any individual said or intended. It states who wrote the formulas down, and when, with sources. That is the whole of its business, and it is enough — a reader who wants to compare a modern retelling against a 499 CE text now has both in one place.
References
Piṅgala, Chandaḥśāstra, c. 3rd–2nd century BCE — mātrā-meru, binary enumeration.
Āryabhaṭa, Āryabhaṭīya, Gaṇitapāda, 499 CE — citighana, the sum of triangular numbers.
Parmanand Singh, “The So-called Fibonacci Numbers in Ancient and Medieval India”, Historia Mathematica 12(3), 229–244 (1985) — the standard account of the Piṅgala → Virahāṅka → Gopāla → Hemachandra chain.
D. M. Burton, Elementary Number Theory, ch. 2 — the Āryabhaṭa attribution as a standard exercise.
Leonardo of Pisa, Liber Abaci, 1202.
This corpus: ch8-meru.html, book4/ch16-crystal-lattice.html, book4/ch21-the-closing-field.html, chSigma-pentanacci.html, chDelta-tetranacci.html, chEta-tribonacci.html.