A227220 -id:A227220 - cp's OEIS frontend

A227218 Smallest triangular number ending in n ones.

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

Offset: 1

Shyam Sunder Gupta, Sep 19 2013

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.

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

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

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

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 *)

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

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

A227219 Smallest triangular number ending in at least n 5's.

Original entry on oeis.org

15, 55, 6555, 2235555, 717655555, 13140555555, 9206385555555, 1551745755555555, 103835341555555555, 26427628585555555555, 2262637355455555555555, 8808604932555555555555, 4704913988655555555555555, 4704913988655555555555555, 19391196055786555555555555555

Offset: 1

Shyam Sunder Gupta, Sep 19 2013

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

			a(3)=6555 because 6555 is the smallest triangular number ending in 3 '5's.

Chai Wah Wu, Table of n, a(n) for n = 1..501

Cf. A000217, A187127, A227218, A227220.

t = {}; Do[Do[x = n*(n + 1)/2;If[Mod[x, 10^m] == 5*(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 + 5*(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 *)

from sympy import sqrt_mod_iter
def A227219(n):
    k, a = 10**n<<1, 10*(10**n-1)//9
    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

a(1) corrected and a(11)-a(15) from Giovanni Resta, Sep 20 2013

Name clarified by Michel Marcus, Sep 05 2019

cp's OEIS Frontend

A227218 Smallest triangular number ending in n ones.

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