A346678 Positive numbers whose squares end in exactly two identical digits.

10, 12, 20, 30, 40, 50, 60, 62, 70, 80, 88, 90, 110, 112, 120, 130, 138, 140, 150, 160, 162, 170, 180, 188, 190, 210, 212, 220, 230, 238, 240, 250, 260, 262, 270, 280, 288, 290, 310, 312, 320, 330, 338, 340, 350, 360, 362, 370, 380, 388, 390, 410, 412, 420, 430, 438, 440, 450, 460

Offset: 1

Author

Bernard Schott, Jul 29 2021

Keywords

• nonn
• base
• easy

Comments

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

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

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)).

Examples

			12 is in the sequence because 12^2 = 144 ends in two 4's.
20 is in the sequence because 20^2 = 400 ends in two 0's.
38 is not in the sequence because 38^2 = 1444 ends in three 4's.
		

Links

• Index entries for linear recurrences with constant coefficients, 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).

Crossrefs

Equals A186438 \ A186439.

Supersequence of A346774.

Programs

Mathematica

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

Python

def ok(n): s = str(n*n); return len(s) > 2 and s[-1] == s[-2] != s[-3]
print(list(filter(ok, range(461)))) # Michael S. Branicky, Jul 29 2021

Formula

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