Triangular numbers

Triangular numbers belong to the oldest surviving strand of Greek arithmetic. The Pythagoreans, working at Croton in southern Italy in the fifth century BCE, arranged pebbles into shapes and read properties off the figures. Their emblem, the tetractys, was the triangle of ten dots produced by 1 + 2 + 3 + 4, and it carried religious weight for them well beyond anything arithmetical. Nicomachus of Gerasa set out the whole family of polygonal numbers systematically in his Introduction to Arithmetic around 100 CE — a textbook that stayed in use for more than a thousand years — and Diophantus of Alexandria wrote a separate treatise On Polygonal Numbers, which survives only as a fragment.

The sequence's most famous modern moment is a story that should be handled with care. Carl Friedrich Gauss is said to have stunned his schoolmaster at Brunswick by adding the integers from 1 to 100 in seconds, having noticed that the numbers pair off into fifty sums of 101. The anecdote is genuinely old: the earliest written version appears in Wolfgang Sartorius von Waltershausen's memorial Gauss zum Gedächtniss of 1856. But that account describes only an arithmetic series, without naming the range 1 to 100, and the vivid details — the teacher, the slate, the exact numbers — vary widely between retellings. The arithmetic is sound and the pairing trick is far older than Gauss; whether the young Gauss performed it exactly as described is not established.

What Gauss demonstrably did came later. On 10 July 1796 he recorded in his diary the line ΕΥΡΗΚΑ! num = Δ + Δ + Δ: a proof that every positive whole number is the sum of at most three triangular numbers. That is the triangular case of a claim Pierre de Fermat had made in 1638 about every polygonal family — that every integer is the sum of at most n n-gonal numbers. Joseph-Louis Lagrange settled the square case in 1770, and Augustin-Louis Cauchy proved the general statement in 1813.

Key properties

• T(n) = n(n+1)/2, so T(1) = 1, T(2) = 3, T(3) = 6, and T(n) = T(n−1) + n.
• T(n) is the sum of the first n positive integers, and equals the binomial coefficient C(n+1, 2).
• T(n) + T(n−1) = n² — two consecutive triangular numbers always add to a perfect square.
• 8·T(n) + 1 = (2n+1)², which gives a quick test: m is triangular exactly when 8m + 1 is a perfect square.
• In base ten a triangular number never ends in 2, 4, 7 or 9; the final digit repeats with period 20.
• Gauss proved in 1796 that every positive integer is the sum of at most three triangular numbers.
• Numbers that are both triangular and square are rare but unlimited in supply: 1, 36, 1225, 41616, 1413721, …
• Every even perfect number is triangular — 6 = T(3), 28 = T(7), 496 = T(31), 8128 = T(127).

Where they turn up

• Ten-pin bowling racks T(4) = 10 pins and a pool table racks T(5) = 15 balls; both are triangles from the same sequence.
• The handshake problem: n people each shaking hands once make T(n−1) handshakes. The same count gives the number of games in a single round-robin tournament and the number of edges in a complete graph on n vertices.
• The third column of Pascal’s triangle is exactly the triangular numbers, because T(n) = C(n+1, 2).
• In "The Twelve Days of Christmas" the gifts delivered on day n total T(n), and the twelve days together come to 364 — one short of a year, which is a coincidence rather than a design.
• Packed storage for a symmetric matrix needs only the lower triangle, so n(n+1)/2 slots instead of n² — a standard space saving in numerical linear algebra libraries.