A227218 Smallest triangular number ending in n ones.

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

Offset

1,2

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.

Links

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

Example

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A352949 A129540 A293480 * A280113 A242102 A221481

Adjacent sequences: A227215 A227216 A227217 * A227219 A227220 A227221

Keyword

nonn,base

Author

Shyam Sunder Gupta, Sep 19 2013

Extensions

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

Status

approved