A227219 -id:A227219 - 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

A227220 Smallest triangular number ending in n 6's.

Original entry on oeis.org

6, 66, 666, 8146666, 2340066666, 109704666666, 15212926666666, 66488766666666, 173147535666666666, 11302559226666666666, 14642337066666666666, 77850029701666666666666, 1029964845546666666666666, 3086210323568466666666666666, 4014564887872666666666666666

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(4)=8146666 because 8146666 is the smallest triangular number ending in  4 '6's.

Robert Israel, Table of n, a(n) for n = 1..500

Cf. A000217, A187127, A227218, A227219.

f:= proc(n) local k;
   k:=min(map(rhs@op, [msolve(k*(k+1)=4/3*(10^n-1),2*10^n)]));
   k*(k+1)/2
end proc:
map(f, [$1..30]); # Robert Israel, Sep 04 2019

t = {}; Do[Do[x = n*(n + 1)/2;If[Mod[x, 10^m] == 6*(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 + 2(10^n - 1)/3 && 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 A227220(n):
    k, a = 10**n<<1, (10**n-1)//3<<2
    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(15) from Giovanni Resta, Sep 20 2013

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A227218 Smallest triangular number ending in n ones.

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