A309597 -id:A309597 - cp's OEIS frontend

A062691 Triangular numbers that contain exactly 2 different digits.

10, 15, 21, 28, 36, 45, 78, 91, 171, 300, 595, 990, 1711, 2211, 3003, 5050, 5151, 5565, 5995, 6555, 8778, 10011, 66066, 222111, 255255, 333336, 500500, 600060, 828828, 887778, 1188111, 5656566, 22221111, 50005000, 51151555, 88877778, 2222211111, 5000050000

Offset: 1

Erich Friedman, Jul 04 2001

For n > 2, A309597(n) is a term. - Seiichi Manyama, Sep 15 2019

The other known infinite families of terms are A037156(n) for n > 1, A319170(n), and A383942(n). - David Radcliffe, Aug 25 2025

			300 is triangular and contains the digits 0 and 3.

David Radcliffe, Table of n, a(n) for n = 1..116 (terms 1..51 from Seiichi Manyama).

Cf. A000217, A045914 (all digits the same), A213516, A213518, A309597.

Select[Accumulate[Range[14000]],Count[DigitCount[#],Except[0]]==2&] (* Harvey P. Dale, Nov 27 2011 *)

for(k=0, 1e5, if(#Set(digits(j=k*(k+1)/2))==2, print1(j", "))) \\ Seiichi Manyama, Sep 15 2019

A213516 Triangular numbers having only 1 or 2 different digits in base 10.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 171, 300, 595, 666, 990, 1711, 2211, 3003, 5050, 5151, 5565, 5995, 6555, 8778, 10011, 66066, 222111, 255255, 333336, 500500, 600060, 828828, 887778, 1188111, 5656566, 22221111, 50005000, 51151555, 88877778

Offset: 1

T. D. Noe, Jun 21 2012

The list of triangular numbers containing only one digit (A045914) is finite. This list is infinite because numbers like 8888777778, 222222111111, and 500000500000 occur an infinite number of times.

A309597 is a subsequence. - Seiichi Manyama, Sep 14 2019

Seiichi Manyama, Table of n, a(n) for n = 1..55

Cf. A000217, A045914, A213517, A309597, A319170.

Cf. A119033 (has list of sequences related to digits in triangular numbers).

[n*(n+1)/2: n in [0..10^5] | #Set(Intseq(n*(n+1) div 2)) le 2]; // Bruno Berselli, Oct 27 2012

t = {}; Do[tri = n*(n+1)/2; If[Length[Union[IntegerDigits[tri]]] <= 2, AppendTo[t, tri]], {n, 0, 10^5}]; t
Select[Accumulate[Range[0,20000]],Count[DigitCount[#],0]>7&] (* Harvey P. Dale, Sep 03 2020 *)

A325907 a(n) = ( (-1)^n * Sum_{k=0..n-2} (-1)^k*10^(2^k) + 10^(2^(n-1)) - ((-1)^n+3)/2 )/3.

Original entry on oeis.org

3, 36, 3363, 33336636, 3333333366663363, 33333333333333336666666633336636, 3333333333333333333333333333333366666666666666663333333366663363

Offset: 1

Seiichi Manyama, Sep 08 2019

nonn

All terms are elements of A213517.

			              36 =        -3 - 1 +        4 * 10^1.
            3363 =       -36 - 1 +       34 * 10^2.
        33336636 =     -3363 - 1 +     3334 * 10^4.
3333333366663363 = -33336636 - 1 + 33333334 * 10^8.
------------------------------------------------------
T(n) = n*(n+1)/2.
               T(3) =                               6.
              T(36) =                             666.
            T(3363) =                         5656566.
        T(33336636) =                 555665666566566.
T(3333333366663363) = 5555555666655656666556566566566.

Seiichi Manyama, Table of n, a(n) for n = 1..10

Cf. A000217, A062691, A093137, A119219, A213517, A309597, A325906.

a[n_] := ((-1)^n * Sum[(-1)^k * 10^(2^k), {k, 0, n - 2}] + 10^(2^(n - 1)) - ((-1)^n + 3)/2)/3; Array[a, 7] (* Amiram Eldar, May 07 2021 *)

{a(n) = ((-1)^n*sum(k=0, n-2, (-1)^k*10^2^k)+10^2^(n-1)-((-1)^n+3)/2)/3}

a(n) = 3 * A325906(n).

a(n) = -a(n-1) - 1 + A093137(2^(n-2)) * 10^(2^(n-2)).

A325493 a(n) = Sum_{k=1..n} A325910(k) * 10^(2^(k-1)).

Original entry on oeis.org

0, 10, 1010, 11011010, 1111001011011010, 11111111000011011111001011011010, 1111111111111111000000001111001011111111000011011111001011011010

Offset: 0

Seiichi Manyama, Sep 14 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..9

Cf. A309597, A325910.

{a(n) = sum(i=1, n, 10^2^(i-1)*((-1)^(i-1)*sum(j=0, i-1, (-1)^j*10^2^j)-(1-(-1)^i)/2)/9)}

A383942 a(n) = (8*10^(2n) - 10^(n+1) + 2) / 9.

Original entry on oeis.org

78, 8778, 887778, 88877778, 8888777778, 888887777778, 88888877777778, 8888888777777778, 888888887777777778, 88888888877777777778, 8888888888777777777778, 888888888887777777777778, 88888888888877777777777778, 8888888888888777777777777778

Offset: 1

David Radcliffe, Aug 18 2025

This is one of four infinite families of triangular numbers consisting of two different digits. The other three families are A319170, A037156 (n>1), and A309597 (n>2).

Paolo Xausa, Table of n, a(n) for n = 1..495
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A000217, A037156, A062691, A073551, A256340, A309597, A319170.

Subsequence of: A119216, A119230, A119236.

A383942[n_] := (8*10^(2*n) - 10^(n+1) + 2)/9; Array[A383942, 15] (* or *)
LinearRecurrence[{111, -1110, 1000}, {78, 8778, 887778}, 15] (* Paolo Xausa, Aug 27 2025 *)

def A383942(n): return (8*10**(2*n)-10**(n+1)+2)//9

a(n) = A000217(A073551(n+1)).

G.f.: 6*x*(13 + 20*x)/((1 - x)*(1 - 10*x)*(1 - 100*x)). - Stefano Spezia, Aug 19 2025

cp's OEIS Frontend

A062691 Triangular numbers that contain exactly 2 different digits.

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A213516 Triangular numbers having only 1 or 2 different digits in base 10.

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A325907 a(n) = ( (-1)^n * Sum_{k=0..n-2} (-1)^k*10^(2^k) + 10^(2^(n-1)) - ((-1)^n+3)/2 )/3.

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A325493 a(n) = Sum_{k=1..n} A325910(k) * 10^(2^(k-1)).

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A383942 a(n) = (8*10^(2n) - 10^(n+1) + 2) / 9.

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