A199691 -id:A199691 - cp's OEIS frontend

A199690 a(n) = (11*10^n + 1)/3.

4, 37, 367, 3667, 36667, 366667, 3666667, 36666667, 366666667, 3666666667, 36666666667, 366666666667, 3666666666667, 36666666666667, 366666666666667, 3666666666666667, 36666666666666667, 366666666666666667, 3666666666666666667, 36666666666666666667, 366666666666666666667

Offset: 0

Vincenzo Librandi, Nov 09 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A199691.

Magma
```
[(11*10^n+1)/3: n in [0..30]];
```

(11*10^Range[0,20]+1)/3 (* or *) LinearRecurrence[{11,-10},{4,37},20] (* Harvey P. Dale, Mar 10 2019 *)

a(n) = 10*a(n-1) - 3.
a(n) = 11*a(n-1) - 10*a(n-2).
G.f.: (4-7*x)/((1-x)*(1-10*x)).
From Elmo R. Oliveira, Jun 10 2025: (Start)
E.g.f.: exp(x)*(1 + 11*exp(9*x))/3.
a(n) = A199691(n)/3. (End)

A379243 a(n) = (10^(n + 1) + 10^(n - min{v_2(n), v_5(n)}) + 1)^n, where v_p(n) indicates the p-adic valuation of n.

Original entry on oeis.org

111, 1212201, 1331363033001, 146415324072600440001, 1610517320513310012100005500001, 1771561966306219615026620001815000066000001, 194871722400927338207105124350046585000254100000770000001, 2143588825589736849603708090188560102487000074536000033880000008800000001

Offset: 1

Marco Ripà, Dec 18 2024

For any n, a(n) == 1 (mod 10^n), while it is not congruent to 1 modulo 10^(n + 1).
If n is not a multiple of 10, 10^(n + 1) + 10^(n - min{v_2(n), v_5(n)}) + 1 has a total of n + 2 digits and they are n - 1 0s and 3 1s. Conversely, if there is a pair of positive integers (m, k) not ending with 0 and such that n := m*10^k, then 10^(n + 1) + 10^(n - min{v_2(n), v_5(n)}) + 1 has n - k + 2 digits (three 1s and the rest are all 0s) and  a(n) = (10^(n + 1) + 10^(n - k) + 1)^n.
Since a(n) == 1 (mod 5) for any n, the constant congruence speed of a(n) (i.e., V(a(n))) is guaranteed to be constant starting from height v_5(a(n) - 1) + 2 (for this sufficient condition, see “Number of stable digits of any integer tetration” in Links).
Then, for any positive integer n, a(n) is (exactly) a n-th perfect power (since, for any given n, 10^(n + 1) + 10^(n - min{v_2(n), v_5(n)}) + 1 is divisible by 3 only once) and is also characterized by a constant congruence speed of n (for a strict proof of the general formula V((10^(t + k) + 10^(t - min{v_2(n), v_5(n)}) + 1)^n) = t, holding for any chosen positive integer k as long as t is an integer above min{v_2(n), v_5(n)} + 1, see Section 3 of “On the relation between perfect powers and tetration frozen digits” in Links).

			a(3) = (10^4 + 10^3 + 1)^3 = 11001^3  = 1331363033001 is a perfect cube whose constant congruence speed is 3.

Marco Ripà and Luca Onnis, Number of stable digits of any integer tetration, Notes on Number Theory and Discrete Mathematics, 2022, 28(3), 441—457.
Marco Ripà, On the relation between perfect powers and tetration frozen digits, Journal of AppliedMath, 2024, 2(5), 1771.
Wikipedia, Tetration.

Cf. A000533, A001597, A063006, A121520, A199691, A317905, A372490, A373387.

pAdicValuation[n_, p_] := Module[{v = 0, k = n}, While[Mod[k, p] == 0 && k > 0,k = k/p;v++;];v];
a[n_] := Module[{v2, v5, minVal}, v2 = pAdicValuation[n, 2]; v5 = pAdicValuation[n, 5];
minVal = Min[v2, v5];(10^(n + 1) + 10^(n - minVal) + 1)^n]; sequence = Table[a[n], {n, 1, 20}]; sequence

If n <> 0 (mod 10), then a(n) = (11...[n - 1 trailing 0s]...1)^n.

cp's OEIS Frontend

A199690 a(n) = (11*10^n + 1)/3.

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