Triangular
One more ball per row, stacked into a number.
The sum of the first n natural numbers, n(n+1)/2 — stackable as a triangle of dots, like 3, 36 or 5050. The rack of billiard balls at break is a triangular number.
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The range holds 1,414 triangular numbers (T₀ through T₁₄₁₃), about 0.14% — and 1,414 happens to be the first digits of √2, a purely beautiful coincidence. Triangular numbers are about as dense as squares, since n(n+1)/2 is essentially half a square.
The classroom legend of young Gauss says it best: told to sum 1+2+…+100 as punishment, he answered 5050 instantly — pair the ends, 50 pairs of 101. That pairing trick is exactly where the triangular formula comes from.
Trivia: Gauss later wrote "Eureka! num = Δ+Δ+Δ" in his diary after proving every positive integer is a sum of at most three triangular numbers. Only five numbers in range are both square and triangular: 0, 1, 36, 1225 and 41616. And 666 is a triangular number (T₃₆) and a palindrome — a double act.