A323639 -id:A323639 - cp's OEIS frontend

A246057 a(n) = (5*10^n - 2)/3.

1, 16, 166, 1666, 16666, 166666, 1666666, 16666666, 166666666, 1666666666, 16666666666, 166666666666, 1666666666666, 16666666666666, 166666666666666, 1666666666666666, 16666666666666666, 166666666666666666, 1666666666666666666, 16666666666666666666, 166666666666666666666

Offset: 0

Vincenzo Librandi, Aug 13 2014

a(k-1) = (10^k - 4)/6, together with b(k) = 3*a(k-1) + 2 = A093143(k) and c(k) = 2*a(k-1) + 1 = A002277(k) are k-digit numbers for k >= 1 satisfying the so-called curious cubic identity a(k-1)^3 + b(k)^3 + c(k)^3 = a(k)*10^(2*k) + b(k)*10^k + c(k) (concatenated a(k)b(k)c(k)). This k-family and the proof of the identity has been given in the introduction of the van der Poorten reference. Thanks go to S. Heinemeyer for bringing these identities to my attention. - Wolfdieter Lang, Feb 07 2017

			Curious cubic identities (see a comment and reference above): 1^3 + 5^3 + 3^3 = 153, 16^3 + 50^3 + 33^3 = 165033, 166^3 + 500^3 + 333^3 = 166500333, ... - _Wolfdieter Lang_, Feb 07 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Alf van der Poorten, Kurt Thomsen, and Mark Wiebe, A curious cubic identity and self-similar sums of squares, The Mathematical Intelligencer, Vol. 29(2), pp. 39-41, March 2007.
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. sequences with terms of the form 1k..k where the digit k is repeated n times: A000042 (k=1), A090843 (k=2), A097166 (k=3), A099914 (k=4), A099915 (k=5), this sequence (k=6), A246058 (k=7), A246059 (k=8), A067272 (k=9).

Cf. A002277, A086948, A093143, A323639.

Magma
```
[(5*10^n-2)/3: n in [0..20]];
```
Mathematica
```
Table[(5 10^n - 2)/3, {n, 0, 20}]
```

vector(50, n, (5*10^(n-1)-2)/3) \\ Derek Orr, Aug 13 2014

G.f.: (1 + 5*x)/((1 - x)*(1 - 10*x)).
a(n) = 11*a(n-1) - 10*a(n-2).
E.g.f.: exp(x)*(5*exp(9*x) - 2)/3. - Stefano Spezia, May 02 2025
a(n) = A323639(n+1)/2 = A086948(n+1)/12. - Elmo R. Oliveira, May 07 2025

A086948 a(n) = k where R(k+8) = 2.

Original entry on oeis.org

12, 192, 1992, 19992, 199992, 1999992, 19999992, 199999992, 1999999992, 19999999992, 199999999992, 1999999999992, 19999999999992, 199999999999992, 1999999999999992, 19999999999999992, 199999999999999992, 1999999999999999992, 19999999999999999992, 199999999999999999992

Offset: 1

Ray Chandler, Jul 24 2003

Vincenzo Librandi, Table of n, a(n) for n = 1..300
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A004086, A085331, A086947, A246057, A323639.

[2*(10^n-4): n in [1..25] ]; // Vincenzo Librandi, Aug 22 2011

Table[10*FromDigits[PadRight[{1},n,9]]+2,{n,20}] (* Harvey P. Dale, Dec 15 2017 *)

a(n) = 2*(10^n - 4).
R(a(n)) = A086947(n).
From Chai Wah Wu, Aug 01 2020: (Start)
a(n) = 11*a(n-1) - 10*a(n-2) for n > 2.
G.f.: x*(60*x + 12)/((x - 1)*(10*x - 1)). (End)
From Elmo R. Oliveira, May 01 2025: (Start)
E.g.f.: 2*(3 - 4*exp(x) + exp(10*x)).
a(n) = 12*A246057(n-1) = 6*A323639(n). (End)

A322571 Positive integers A270710(k) (= (k+1)(3k-1)) which have only 1 or 2 different digits in base 10.

Original entry on oeis.org

4, 15, 32, 55, 84, 119, 455, 799, 900, 3332, 3535, 7007, 8855, 244244, 333332, 335335, 400404, 445444, 555559, 666464, 799799, 1999199, 3303300, 33333332, 33353335, 3333333332, 3333533335, 333333333332, 333335333335, 700007077007, 33333333333332, 33333353333335, 3333333333333332, 3333333533333335

Offset: 1

Seiichi Manyama, Aug 29 2019

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

Cf. A002277, A018885 (in case of squares), A213516 (in case of triangular numbers), A270710, A322570, A323639.

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

Select[Table[(n+1)(3n-1),{n,3334*10^4}],Count[DigitCount[#],0]>7&] (* Harvey P. Dale, Jun 12 2022 *)

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

a(n) = A270710(A322570(n)).

For k > 0, A002277(2*k) - 1 is a term.

cp's OEIS Frontend

A246057 a(n) = (5*10^n - 2)/3.

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A086948 a(n) = k where R(k+8) = 2.

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A322571 Positive integers A270710(k) (= (k+1)(3k-1)) which have only 1 or 2 different digits in base 10.

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cp's OEIS Frontend

A246057 a(n) = (5*10^n - 2)/3.

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Author

Keywords

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Examples

Links

Crossrefs

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Magma

Mathematica

PARI

Formula

A086948 a(n) = k where R(k+8) = 2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A322571 Positive integers A270710(k) (= (k+1)*(3*k-1)) which have only 1 or 2 different digits in base 10.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A322571 Positive integers A270710(k) (= (k+1)(3k-1)) which have only 1 or 2 different digits in base 10.