FIGURATE NUMBERS VISUALIZER
SHAPE
Triangular
Square
Pentagonal
Hexagonal
Star
Tetrahedral
Cubic
Octahedral
N VALUE
n =
8
Animate
FORMULA
SEQUENCE
IDENTITIES
T(n) + T(n-1) = n²
T(n) = C(n+1,2)
8T(n)+1 = (2n+1)²
T(m)+T(n)+T(k) covers all
(Gauss "Eureka" 1796)
Every integer is sum of
≤3 triangular numbers,
≤4 squares (Lagrange),
≤5 pentagonals.