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A226772 Triangular numbers obtained as the concatenation of n and 2n. 5
36, 1326, 2346, 3570, 125250, 223446, 12502500, 22234446, 1250025000, 2066441328, 2222344446, 2383847676, 3673573470, 125000250000, 222223444446, 5794481158896, 12500002500000, 12857132571426, 22222234444446, 49293309858660, 804878916097578, 933618918672378, 971908519438170 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

LINKS

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

EXAMPLE

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

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 *)

PROG

(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)))}

CROSSREFS

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

Sequence in context: A209042 A283729 A203333 * A158739 A218518 A099366

Adjacent sequences:  A226769 A226770 A226771 * A226773 A226774 A226775

KEYWORD

nonn,base

AUTHOR

Antonio Roldán, Jun 18 2013

STATUS

approved

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Last modified March 7 19:49 EST 2021. Contains 341931 sequences. (Running on oeis4.)