A227218 Smallest triangular number ending in n ones.

1, 1711, 105111, 6271111, 664611111, 222222111111, 22222221111111, 2222222211111111, 2517912111111111, 18428299161111111111, 2222222222211111111111, 222222222222111111111111, 22222222222221111111111111, 1090161504430911111111111111

Offset: 1

Author

Shyam Sunder Gupta, Sep 19 2013

Keywords

• nonn
• base

Comments

If a triangular number ends in like digits then it can only end in 0's, 1's, 5's or 6's.

The sequence is infinite because the sequence of triangular numbers 21, 2211, 222111, 22221111, ... (A319170) is infinite.

Examples

			a(2) = 1711 because 1711 is the smallest triangular number ending in  2 '1's.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..501 (terms 1..200 from Giovanni Resta)

Crossrefs

Cf. A000217, A187127, A227219, A227220, A319170.

Programs

Mathematica

t = {}; Do[Do[x = n*(n + 1)/2;If[Mod[x, 10^m] == (10^m - 1)/9, AppendTo[t, x]; Break[]], {n, 1, 10^m}], {m, 1, 10}]; t
a[n_] := Block[{x, y, s}, s = y /. List@ ToRules[ Reduce[(y+1)* y/2 == x*10^n +(10^n - 1)/9 && y > 0 && x >= 0, {y, x}, Integers] /. C[1] -> 0]; Min[s*(s + 1)/2]]; Array[a, 20] (* Giovanni Resta, Sep 20 2013 *)

Python

from sympy import sqrt_mod_iter
def A227218(n):
    k, a = 10**n<<1, (10**n-1)//9<<1
    m = (a<<2)+1
    return min(b for b in ((d>>1)*((d>>1)+1) for d in sqrt_mod_iter(m, k) if d&1) if b%k==a)>>1 # Chai Wah Wu, May 04 2024

Extensions

a(11)-a(14) from Giovanni Resta, Sep 20 2013