A226742 Triangular numbers obtained as the concatenation of 2*k and k.

21, 105, 2211, 9045, 222111, 306153, 742371, 890445, 1050525, 22221111, 88904445, 107905395, 173808690, 2222211111, 8889044445, 12141260706, 15754278771, 222222111111, 888890444445, 22222221111111, 36734701836735, 65306123265306

Offset: 1

Author

Antonio Roldán, Jun 18 2013

Keywords

• nonn
• base

Comments

Includes (2*10^k+1)*(10^k-1)/9 and (2*10^k+1)*(4*10^k+5)/9 for k >= 1. - Robert Israel, Feb 06 2025

Examples

			If k=111, 2k=222, 2k//k = 222111 = 666*667/2, a triangular number.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..2675

Crossrefs

Cf. A003098, A068899, A226772, A226788, A226789, A380792.

Programs

Maple

g:= proc(d) local a, b, n, Res, x, y;
      Res:= NULL:
      for a in numtheory:-divisors(2*(2*10^d+1)) do
        b:= 2*(2*10^d+1)/a;
        if igcd(a, b)>1 then next fi;
        n:= chrem([0, -1], [a, b]);
        x:= n*(n+1)/2;
        y:= x/(2*10^d+1);
        if y < 10^(d-1) or y >= 10^d  then next fi;
        Res:= Res, (2*10^d+1)*y
      od;
      op(sort([Res]))
end proc:
map(g, [$1..10]); # Robert Israel, Feb 06 2025

Mathematica

TriangularQ[n_] := IntegerQ[Sqrt[1 + 8*n]]; t = {}; Do[s = FromDigits[Join[IntegerDigits[2*n], IntegerDigits[n]]]; If[TriangularQ[s], AppendTo[t, s]], {n, 100000}]; t (* T. D. Noe, Jun 18 2013 *)

PARI

concatint(a,b)=eval(concat(Str(a),Str(b)))
istriang(x)=issquare(8*x+1)
{for(n=1,10^5,a=concatint(2*n,n);if(istriang(a),print(a)))}