This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A353139 #15 Apr 28 2022 07:48:58
%S A353139 212,781,794,806,817,838,841,844,865,2962,3101,3130,3171,3178,3185,
%T A353139 3213,3219,3226,3269,3274,3335,3353,3354,3356,3370,3378,3490,3496,
%U A353139 3521,3528,3595,3597,3606,3610,3626,3651,3672,3718,3777,11797,11798,11850,11938,12049
%N A353139 Digitally balanced numbers (A031443) whose squares are also digitally balanced.
%C A353139 Numbers x such that both x and x^2 are terms of A031443, that is, have the same number of 0's as 1's in their binary representations.
%t A353139 balQ[n_] := Module[{d = IntegerDigits[n, 2], m}, EvenQ @ (m = Length @ d) && Count[d, 1] == m/2]; Select[Range[12000], balQ[#] && balQ[#^2] &] (* _Amiram Eldar_, Apr 26 2022 *)
%o A353139 (Python)
%o A353139 from itertools import count, islice
%o A353139 from sympy.utilities.iterables import multiset_permutations
%o A353139 def isbalanced(n): b = bin(n)[2:]; return b.count("0") == b.count("1")
%o A353139 def A031443gen(): yield from (int("1"+"".join(p), 2) for n in count(1) for p in multiset_permutations("0"*n+"1"*(n-1)))
%o A353139 def agen():
%o A353139 for k in A031443gen():
%o A353139 if isbalanced(k**2):
%o A353139 yield k
%o A353139 print(list(islice(agen(), 40))) # _Michael S. Branicky_, Apr 26 2022
%Y A353139 Cf. A031443, A345397.
%K A353139 nonn,base
%O A353139 1,1
%A A353139 _Alex Ratushnyak_, Apr 26 2022