A186438 Positive numbers whose squares end in two identical digits.

10, 12, 20, 30, 38, 40, 50, 60, 62, 70, 80, 88, 90, 100, 110, 112, 120, 130, 138, 140, 150, 160, 162, 170, 180, 188, 190, 200, 210, 212, 220, 230, 238, 240, 250, 260, 262, 270, 280, 288, 290, 300, 310, 312, 320, 330, 338, 340, 350, 360, 362, 370, 380, 388, 390, 400, 410, 412

Offset

1,1

Comments

The numbers are of the form : 10k, or 50k - 12, or 50k + 12, or 50k + 38.

References

Jean Meeus, Letter to N. J. A. Sloane, Dec 26 1974.

Links

Table of n, a(n) for n=1..58.

J. Meeus & N. J. A. Sloane, Correspondence, 1974-1975

Ana Rechtman, Juillet 2021, 4e défi, Images des Mathématiques, CNRS, 2021 (in French).

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).

Formula

G.f.: 2*x*(5*x^6+x^5+4*x^4+5*x^3+4*x^2+x+5)/((x-1)^2 * (x^6+x^5+x^4+x^3+x^2+x+1)). [Colin Barker, Jul 02 2012]

Example

62 is in the sequence because 62^2 = 3844.

Maple

with(numtheory):T:=array(1..10):for p from 1 to 1000 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]
  then printf(`%d, `, p):else fi:od:

Mathematica

tidQ[n_]:=Module[{idn=IntegerDigits[n^2]}, idn[[-1]]==idn[[-2]]]; Select[ Range[ 4, 500], tidQ] (* or *) LinearRecurrence[{1, 0, 0, 0, 0, 0, 1, -1}, {10, 12, 20, 30, 38, 40, 50, 60}, 60] (* Harvey P. Dale, Jan 25 2014 *)

Crossrefs

Cf. A016742 (even squares), A123912.

Sequence in context: A358998 A085772 A162189 * A346678 A035284 A265403

Adjacent sequences: A186435 A186436 A186437 * A186439 A186440 A186441

Keyword

nonn,base,easy,changed

Author

Michel Lagneau, Feb 21 2011

Status

approved