Squares that are the sum of two repdigit numbers.
0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 225, 784, 1444, 10000, 44521, 1000000, 100000000, 10000000000, 1000000000000, 100000000000000, 10000000000000000, 1000000000000000000, 100000000000000000000, 10000000000000000000000
1
A repdigit number is a number whose digits are all the same, for example, 11111. If we ignore the powers are 10, then this sequence is finite with last term 44521 = 211^2 = 77 + 44444. For completeness, we allow 0 as a repdigit number.
T. D. Noe, Plot of 27 terms
Bart Goddard and Jeremy Rouse, Sum of two repdigits a square, arXiv 1607.06681 (Jul 22 2016)
(Mma) t = Sort[Flatten[Table[FromDigits[Table[a, {len}]], {a, 0, 9}, {len, 22}]]]; t2 = Sort[Flatten[Outer[Plus, t, t]]]; Union[Select[t2, IntegerQ[Sqrt[#]] &]]
Cf. A010785 (repdigit numbers).
nonn,base
T. D. Noe, Jul 25 2016