A186439 Numbers whose squares end in three identical digits.

38, 100, 200, 300, 400, 462, 500, 538, 600, 700, 800, 900, 962, 1000, 1038, 1100, 1200, 1300, 1400, 1462, 1500, 1538, 1600, 1700, 1800, 1900, 1962, 2000, 2038, 2100, 2200, 2300, 2400, 2462, 2500, 2538, 2600, 2700, 2800, 2900, 2962, 3000, 3038, 3100, 3200, 3300, 3400, 3462

Offset: 1

Author

Michel Lagneau, Feb 21 2011

Keywords

• nonn
• base
• easy

Comments

The three ending digits of a(n)^2 are 000 or 444.

n is in the sequence iff either n == 0 mod 100 or n == (+/-)38 mod 500. - Robert Israel, Jul 03 2014

Examples

			462 is in the sequence because 462^2 = 213444.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).

Crossrefs

Cf. A016742 (even squares), A186438.

Cf. A039685, A047336.

Cf. A346678.

Programs

Maple

with(numtheory):T:=array(1..10):for p from 1 to 10000 do:n:=p^2:l:=length(n):n0:=n:for
  m from 1 to l do:q:=n0:u:=irem(q,10):v:=iquo(q,10):n0:=v :T[m]:=u:od:if T[1]=T[2]
  and T[1]=T[3] then printf(`%d, `,p):else fi:od:
# second Maple program:
a:= proc(n) local m, r;
      r:= 1+ irem(n-1, 7, 'm');
      [38, 100, 200, 300, 400, 462, 500][r] +500*m
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 24 2011

Mathematica

Select[Range[11,10000],Mod[PowerMod[#,2,1000],111]==0&] (* Zak Seidov, Feb 23 2011 *)

PARI

for(n=11,10000,if((n^2%1000)%111==0,print1(n", "))) \\ Zak Seidov, Feb 23 2011

PARI

Vec(2*x*(19*x^2 +12*x +19)*(x^4 +x^3 +x^2 +x +1)/((x -1)^2*(x^6 +x^5 +x^4 +x^3 +x^2 +x +1)) + O(x^100)) \\ Colin Barker, Jul 03 2014

Python

def ok(n): s = str(n*n); return len(s) > 2 and s[-1] == s[-2] == s[-3]
print(list(filter(ok, range(3463)))) # Michael S. Branicky, Jul 29 2021

Formula

a(A047336(n)) = A039685(n). - Bruno Berselli, Feb 22 2011

a(n) = a(n-7) + 500 for n > 7. - Zak Seidov and Bruno Berselli, Feb 23 2011

G.f.: 2*x*(19*x^2 +12*x +19)*(x^4 +x^3 +x^2 +x +1) / ((x -1)^2*(x^6 +x^5 +x^4 +x^3 +x^2 +x +1)). - Colin Barker, Jul 03 2014