[seqfan] Patterns in A052061?

[zak seidov]

Dear seqfans,

Prove or disprove that
numbers n with one '3' and arbitrary numbers of '1's: 13, 113, 1113,... 
are such that decimal expansion of n^2 contains no palindromic  substring except single digits
(and hence all such n's are in  A052061):
{13,169}
{113,12769}
{1113,1238769}
{11113,123498769}
{111113,12346098769}
{1111113,1234572098769}
{11111113,123456832098769}
{111111113,12345679432098769}
{1111111113,1234567905432098769}
{11111111113,123456790165432098769}.

Other similar patterns?

Thanks, 
Zak