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A186439 Numbers whose squares end in three identical digits.

%I A186439 #45 Jul 29 2021 07:28:00
%S A186439 38,100,200,300,400,462,500,538,600,700,800,900,962,1000,1038,1100,
%T A186439 1200,1300,1400,1462,1500,1538,1600,1700,1800,1900,1962,2000,2038,
%U A186439 2100,2200,2300,2400,2462,2500,2538,2600,2700,2800,2900,2962,3000,3038,3100,3200,3300,3400,3462
%N A186439 Numbers whose squares end in three identical digits.
%C A186439 The three ending digits of a(n)^2 are 000 or 444.
%C A186439 n is in the sequence iff either n == 0 mod 100 or n == (+/-)38 mod 500. - _Robert Israel_, Jul 03 2014
%H A186439 Alois P. Heinz, <a href="/A186439/b186439.txt">Table of n, a(n) for n = 1..1000</a>
%H A186439 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,1,-1).
%F A186439 a(A047336(n)) = A039685(n). - _Bruno Berselli_, Feb 22 2011
%F A186439 a(n) = a(n-7) + 500 for n > 7. - _Zak Seidov_ and _Bruno Berselli_, Feb 23 2011
%F A186439 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
%e A186439 462 is in the sequence because 462^2 = 213444.
%p A186439 with(numtheory):T:=array(1..10):for p from 1 to 10000 do:n:=p^2:l:=length(n):n0:=n:for
%p A186439   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]
%p A186439   and T[1]=T[3] then printf(`%d, `,p):else fi:od:
%p A186439 # second Maple program:
%p A186439 a:= proc(n) local m, r;
%p A186439       r:= 1+ irem(n-1, 7, 'm');
%p A186439       [38, 100, 200, 300, 400, 462, 500][r] +500*m
%p A186439     end:
%p A186439 seq(a(n), n=1..100);  # _Alois P. Heinz_, Feb 24 2011
%t A186439 Select[Range[11,10000],Mod[PowerMod[#,2,1000],111]==0&] (* _Zak Seidov_, Feb 23 2011 *)
%o A186439 (PARI) for(n=11,10000,if((n^2%1000)%111==0,print1(n", "))) \\ _Zak Seidov_, Feb 23 2011
%o A186439 (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
%o A186439 (Python)
%o A186439 def ok(n): s = str(n*n); return len(s) > 2 and s[-1] == s[-2] == s[-3]
%o A186439 print(list(filter(ok, range(3463)))) # _Michael S. Branicky_, Jul 29 2021
%Y A186439 Cf. A016742 (even squares), A186438.
%Y A186439 Cf. A039685, A047336.
%Y A186439 Cf. A346678.
%K A186439 nonn,base,easy
%O A186439 1,1
%A A186439 _Michel Lagneau_, Feb 21 2011