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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. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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
R. Israel, Re: Help with a(n) and a cumulative sum, Seqfan (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
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 *)
PROG
(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
(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, Ul 20 2019
CROSSREFS
Cf. A289404.
Sequence in context: A260716 A112521 A192350 * A354949 A239861 A266850
KEYWORD
nonn,base,easy
AUTHOR
N. J. A. Sloane, Jul 15 2019
STATUS
approved

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Last modified May 24 22:09 EDT 2024. Contains 372782 sequences. (Running on oeis4.)