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

1,1

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

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).

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A016742 (even squares), A186438.

Cf. A039685, A047336.

Cf. A346678.

Sequence in context: A124141 A093649 A020167 * A044225 A044606 A244177

Adjacent sequences: A186436 A186437 A186438 * A186440 A186441 A186442

Keyword

nonn,base,easy

Author

Michel Lagneau, Feb 21 2011

Status

approved