Concept
Triangular numbers
The counts of dots in a triangular arrangement - 1, 3, 6, 10 and onward - each the running total of the whole numbers up to its own index. They are the third diagonal of Pascal's triangle, because a diagonal there is the running total of the one before it, and twice one of them is a rectangle.
Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.
Every square is a stack of odd numbers
Add up the odd numbers in order and the running totals are 1, 4, 9, 16, 25. This is not a coincidence, and the reason fits in a single picture.
The run that lands one place along
Add up a run of entries down one of Pascal's diagonals and the total is another entry of the triangle — one row further down and one place along. The same triangle holds four more sums of that kind, and each is a different question answered by the same additive rule.
Named alongside it
The objects these essays reach for when they reach for this one.
Counting two waysAlgebraic identityBinomial coefficientCombinatorial proofFibonacciGnomonRecursionSquare numbersTelescopingUniform acceleration