A308900 An explicit example of an infinite sequence with a(1)=1 and, for n >= 2, a(n) and S(n) = Sum_{i=1..n} a(i) have no digit in common.

1, 6, 4, 66, 34, 666, 334, 6666, 3334, 66666, 33334, 666666, 333334, 6666666, 3333334, 66666666, 33333334, 666666666, 333333334, 6666666666, 3333333334, 66666666666, 33333333334, 666666666666, 333333333334, 6666666666666, 3333333333334, 66666666666666, 33333333333334

Offset: 1

Author

N. J. A. Sloane, Jul 15 2019

Keywords

• nonn
• base
• easy

Comments

Used in a proof that the initial terms of A309151 are correct.

The S(n) sequence is 1, 7, 11, 77, 111, 777, 1111, 7777, 11111, 77777, ...

A093137 interleaved with positive terms of A002280. - Felix Fröhlich, Jul 15 2019

Links

• Robert Israel, Table of n, a(n) for n = 1..1999
• R. Israel, Re: Help with a(n) and a cumulative sum, Seqfan (Jul 15 2019).
• Index entries for linear recurrences with constant coefficients, signature (-1,10,10).

Crossrefs

Cf. A002280, A093137, A309151.

Cf. A289404.

Programs

Magma

I:=[1,6,4]; [n le 3 select I[n] else - Self(n-1) + 10*Self(n-2) + 10*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jul 20 2019

Maple

1, seq(op([6*(10^i-1)/9, 3*(10^i-1)/9+1]), i=1..30); # Robert Israel, Jul 15 2019

Mathematica

CoefficientList[Series[(1 + 7 x)/((1 + x) (1 - 10 x^2)), {x, 0, 26}], x] (* Michael De Vlieger, Jul 18 2019 *)
LinearRecurrence[{-1,10,10},{1,6,4},30] (* Harvey P. Dale, Jan 02 2022 *)

PARI

Vec((1+7*x)/((1+x)*(1-10*x^2)) + O(x^20)) \\ Felix Fröhlich, Jul 15 2019

PARI

a(n) = if(n==1, 1, if(n%2==0, 6*(10^(n/2)-1)/9, 3*(10^((n-1)/2)-1)/9+1)) \\ Felix Fröhlich, Jul 15 2019

Formula

For even n >= 2, a(n) = 6666...66 (with n/2 6's). For odd n >= 5, a(n) = 3333...334 (with (n-3)/2 3's and a single 4).

From Robert Israel, Jul 15 2019: (Start)

G.f. (1+7*x)/((1+x)*(1-10*x^2)).

a(n) = -a(n - 1) + 10*a(n - 2) + 10*a(n - 3). (End)

a(-n) = a(n+1). - Paul Curtz, Jul 18 2019

a(n) = (1/60)*(-40*(-1)^n + (1 + (-1)^n)*(2^(2+n/2)*5^(1+n/2)) + (1 + (-1)^(n+1))*10^((1+n)/2)). - Stefano Spezia, Jul 20 2019