153 as a number between squares

153 can be expressed as a² + b² and as (a+1)² – (b+1)².  In other words, 12² + 3² = 153 = 13² – 4².

This feature applies for every natural number b, with a equal to  b²+b.  So if b=4, this means a = 16 + 4 = 20.  This gives 416 = 400 + 16 = 441 – 25.

The first 10 numbers with this feature are 5  40  153  416  925  1,720  3,185  5,248  8,181 12,200.

What’s more unusual with 153 is that b² + (b+1)² is also a square: 3² + 4² = 5².  The next b where this applies is 20 (20² + 21² = 29²).  In that case, a = 420.  this gives a² + b² = 176,800.  After that the next is b = 119, a = 14,280 – that gives a² + b² = 203,932,561.

So 153 having this feature makes it part of a very select group.