A213517 -id:A213517 - cp's OEIS frontend

A118668 Number of distinct digits needed to write the n-th triangular number in decimal representation.

1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 1, 3, 3, 3, 3, 3, 3, 3, 2, 4, 3, 3, 3, 3, 4, 4, 4, 3, 4, 4, 4, 4, 2, 3, 4, 3, 4, 4, 3, 4, 2, 3, 4, 4, 4, 3, 3, 4, 3, 4, 3, 2, 4, 4, 4, 3, 3, 4, 4, 3, 4, 3, 4, 3, 4, 4, 4, 4, 3, 4, 3, 4, 4, 4, 2, 2, 3, 3, 4

Offset: 0

Reinhard Zumkeller, May 19 2006

0 < a(n) <= 10;

a(n) = A043537(A000217(n)).

			n=99: 99*(99+1)/2 = 4950 -> a(99) = #{0,4,5,9} = 4;
see A119033 for an overview of sequences with terms composed of not more than 3 distinct digits.
n=100: 100*(100+1)/2 = 5050 -> a(100) = #{0,5} = 2;

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A000217, A043537, A045914, A213517, A213518, A255678.

a118668 = a043537 . a000217
a118668_list = map a043537 a000217_list
-- Reinhard Zumkeller, Jul 11 2015

Length[Union[IntegerDigits[#]]]&/@Accumulate[Range[0,110]] (* Harvey P. Dale, Jul 23 2012 *)

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)).

A213518 Numbers k such that the triangular number k*(k+1)/2 has 2 different digits in base 10.

Original entry on oeis.org

4, 5, 6, 7, 8, 9, 12, 13, 18, 24, 34, 44, 58, 66, 77, 100, 101, 105, 109, 114, 132, 141, 363, 666, 714, 816, 1000, 1095, 1287, 1332, 1541, 3363, 6666, 10000, 10114, 13332, 66666, 100000, 133332, 666666, 1000000, 1333332, 6666666, 10000000, 13333332, 33336636, 66666666, 100000000

Offset: 1

T. D. Noe, Jun 22 2012

The list of triangular numbers containing only one digit (A045914) is finite. This list is infinite because numbers like 133332, 666666, and 1000000 occur an infinite number of times.
A118668(a(n)) = 2. - Reinhard Zumkeller, Jul 11 2015
For n > 2, A325907(n) is a term. - Seiichi Manyama, Sep 15 2019

David Radcliffe, Table of n, a(n) for n = 1..116 (terms 1..51 from Seiichi Manyama, terms 52..60 from T. D. Noe).

Cf. A062691 (the corresponding triangular numbers), A213516, A213517, A325907.
Cf. A118668.
Cf. A187127.

a213518 n = a213518_list !! (n-1)
a213518_list = filter ((== 2) . a118668) [0..]
-- Reinhard Zumkeller, Jul 11 2015

t = {}; Do[tri = n*(n+1)/2; If[Length[Union[IntegerDigits[tri]]] == 2, AppendTo[t, n]], {n, 10^5}]; t

for(k=0, 1e8, if(#Set(digits(k*(k+1)/2))==2, print1(k", "))) \\ Seiichi Manyama, Sep 15 2019

a(45)-a(48) from Seiichi Manyama, Sep 15 2019

A322570 Positive integers k such that A270710(k) (= (k+1)(3k-1)) have only 1 or 2 different digits in base 10.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 12, 16, 17, 33, 34, 48, 54, 285, 333, 334, 365, 385, 430, 471, 516, 816, 1049, 3333, 3334, 33333, 33334, 333333, 333334, 483048, 3333333, 3333334, 33333333, 33333334, 333333333, 333333334, 3333333333, 3333333334, 33333333333, 33333333334

Offset: 1

Seiichi Manyama, Aug 29 2019

Giovanni Resta, Table of n, a(n) for n = 1..50

Cf. A002277, A016069, A093137, A213517 (in case of triangular numbers), A270710, A322571.

[k:k in [1..10000000]| #Set(Intseq((k+1)*(3*k-1))) le 2]; // Marius A. Burtea, Aug 29 2019

Select[Range@ 50000, Length@ Union@ IntegerDigits[3 #^2 + 2 # - 1] <= 2 &] (* Giovanni Resta, Sep 04 2019 *)

for(k=1, 1e8, if(#Set(digits(3*k^2+2*k-1))<=2, print1(k", ")))

For k > 0, A002277(k) is a term.

For k >= 0, A002277(k) + 1 (= A093137(k)) is a term.

a(35)-a(36) from Jinyuan Wang, Aug 30 2019

a(37)-a(40) from Giovanni Resta, Sep 04 2019

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A118668 Number of distinct digits needed to write the n-th triangular number in decimal representation.

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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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A213518 Numbers k such that the triangular number k*(k+1)/2 has 2 different digits in base 10.

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A322570 Positive integers k such that A270710(k) (= (k+1)*(3*k-1)) have only 1 or 2 different digits in base 10.

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A322570 Positive integers k such that A270710(k) (= (k+1)(3k-1)) have only 1 or 2 different digits in base 10.