Multimagic square 16, 25, 36, 49, ... Sum of Squares 0, 1, 5, 14, 30, 55, 91, 140, 204, 285, ... )number of units in a square-based pyramid) The bimagic series is the 1st, 4th, 9th in this series (divided by 1, 2, 3, n) etc. so values for the rows and columns in order-1, order-2, order-3 Bimagic squares would be 1, 15, 95, 374, 1105, 2701, 5775, 11180, ... The trimagic series would be related in the same way to the hyper-pyramidal sequence of nested cubes. Cubes 0, 1, 8, 27, 64, 125, 216, ... Sum of Cubes 0, 1, 9,
1 + 2 + 4 + 8 + ⋯ that and , but the underlying proofs demand careful thinking about the interpretation of endless sums. It is also possible to view this series as convergent in a number system different from the real numbers, namely, the 2-adic numbers. As a series of 2-adic numbers this series converges to the same sum, −1, as was derived above by analytic continuation. 1 + 2 + 4 + 8 + ⋯ In mathematics, is the infinite series whose terms are the successive powers of two. As a geometric series, it is characterized by its first term, 1, and its common ratio, 2.