A226772 Triangular numbers obtained as the concatenation of n and 2n.

36, 1326, 2346, 3570, 125250, 223446, 12502500, 22234446, 1250025000, 2066441328, 2222344446, 2383847676, 3673573470, 125000250000, 222223444446, 5794481158896, 12500002500000, 12857132571426, 22222234444446, 49293309858660, 804878916097578, 933618918672378, 971908519438170

Offset: 1

Author

Antonio Roldán, Jun 18 2013

Keywords

• nonn
• base

Comments

Includes 125*10^(2*k+1)+25*10^k and (10^k+2)*(1+(10^k-1)*2/9) for k >= 1. - Robert Israel, Nov 09 2020

Examples

			If n=23, 2n=46, n//2n = 2346 = 68*69/2, a triangular number.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..1509

Crossrefs

Cf. A003098, A068899, A226742, A226788, A226789.

Programs

Maple

F:= proc(d) local D,R,M,m,w,x,x1,x2;
   R:= NULL;
   M:= 10^d/2+1;
   D:= numtheory:-divisors(M);
   for m in D do if igcd(m,M/m)=1 then
     for w in [chrem([-1,1],[8*m,M/m]), chrem([1,-1],[8*m,M/m])] do
     x:= (w^2-1)/8;
     x1:= x mod 10^d;
     x2:= floor(x/10^d);
     if x1 = 2*x2 and x1 >= 10^(d-1) then R:= R, x fi
   od fi od;
   op(sort([R]))
end proc:
36, seq(F(d),d=2..10); # Robert Israel, Nov 09 2020

Mathematica

TriangularQ[n_] := IntegerQ[Sqrt[1 + 8*n]]; t = {}; Do[s = FromDigits[Join[IntegerDigits[n], IntegerDigits[2*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(n,2*n);if(istriang(a),print(a)))}