A centered triangular number has a dot at the centre surrounded by triangular rings: 1, 4, 10, 19, 31, 46, 64, 85, 109. The formula is \((3k^2 - 3k + 2)/2\).
They complete, with the [[centered-square]] and [[centered-hexagonal]] numbers, the family of centered polygonal numbers — distinct from the ordinary [[triangular]] numbers, which build from a corner rather than a centre. Every centered triangular number leaves remainder 1 when divided by 3.