%I #35 Aug 05 2021 17:08:33

%S 10,12,20,30,40,50,60,62,70,80,88,90,110,112,120,130,138,140,150,160,

%T 162,170,180,188,190,210,212,220,230,238,240,250,260,262,270,280,288,

%U 290,310,312,320,330,338,340,350,360,362,370,380,388,390,410,412,420,430,438,440,450,460

%N Positive numbers whose squares end in exactly two identical digits.

%C When a square ends in exactly two identical digits, these digits are necessarily 00 or 44, so all terms are even.

%C The numbers are of the form: 10*floor((10*k-1)/9), k > 0, or, 50*floor((10*k-1)/9) +- 38, k > 0.

%C Equivalently: m is in the sequence iff either (m == 0 (mod 10) and m <> 0 (mod 100)) or (m == +- 38 (mod 50) and m <> +- 38 (mod 500)).

%H <a href="/index/Rec#order_64">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1).

%F a(n+63) = a(n) + 500.

%e 12 is in the sequence because 12^2 = 144 ends in two 4's.

%e 20 is in the sequence because 20^2 = 400 ends in two 0's.

%e 38 is not in the sequence because 38^2 = 1444 ends in three 4's.

%t Select[Range[10, 460], (d = IntegerDigits[#^2])[[-1]] == d[[-2]] != d[[-3]] &] (* _Amiram Eldar_, Jul 29 2021 *)

%o (Python)

%o def ok(n): s = str(n*n); return len(s) > 2 and s[-1] == s[-2] != s[-3]

%o print(list(filter(ok, range(461)))) # _Michael S. Branicky_, Jul 29 2021

%Y Equals A186438 \ A186439.

%Y Supersequence of A346774.

%K nonn,base,easy

%O 1,1

%A _Bernard Schott_, Jul 29 2021