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

Author

Michel Lagneau, Feb 21 2011

Keywords

• nonn
• base
• easy

Comments

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

Examples

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

References

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

Links

• 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).

Crossrefs

Cf. A016742 (even squares), A123912.

Programs

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 *)

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]