A068899 Triangular numbers containing 2n digits obtained by duplicating the first n digits; i.e., triangular numbers in A020338.

55, 66, 5050, 5151, 203203, 255255, 426426, 500500, 501501, 581581, 828828, 930930, 39653965, 50005000, 50015001, 61566156, 3347133471, 5000050000, 5000150001, 6983669836, 220028220028, 500000500000, 500001500001

Offset: 1

Author

Amarnath Murthy, Mar 21 2002

Keywords

• easy
• nonn
• base
• look

Comments

The sequence is infinite: the 10^n-th and the (10^n + 1)-th triangular numbers are members. It is a subsequence of A068898.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A020338, A068898.

Programs

Maple

N:= 10: # to get all terms of up to 2N digits
Res:= NULL:
for n from 1 to N do
   Divs:= select(t -> igcd(t,(10^n+1)/t)=1, numtheory:-divisors(10^n+1));
   for d in Divs do
     for e in [1,3] do
      u:= chrem([1,-1,e],[d,(10^n+1)/d,4]);
      y:= (u^2-1)/8/(10^n+1);
      if y >= 10^(n-1) and y < 10^n then Res:= Res, y*(10^n+1) fi;
od od od:
sort([Res]); # Robert Israel, Feb 27 2017

Mathematica

Select[Accumulate[Range[5*10^6]],EvenQ[IntegerLength[#]]&&Take[ IntegerDigits[ #],IntegerLength[ #]/2]== Take[IntegerDigits[#],-IntegerLength[#]/2]&] (* Harvey P. Dale, Aug 20 2022 *)

Extensions

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jan 10 2003