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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.

%I A308900 #45 Aug 04 2025 01:07:02
%S A308900 1,6,4,66,34,666,334,6666,3334,66666,33334,666666,333334,6666666,
%T A308900 3333334,66666666,33333334,666666666,333333334,6666666666,3333333334,
%U A308900 66666666666,33333333334,666666666666,333333333334,6666666666666,3333333333334,66666666666666,33333333333334
%N 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.
%C A308900 Used in a proof that the initial terms of A309151 are correct.
%C A308900 The S(n) sequence is 1, 7, 11, 77, 111, 777, 1111, 7777, 11111, 77777, ...
%C A308900 A093137 interleaved with positive terms of A002280. - _Felix Fröhlich_, Jul 15 2019
%H A308900 Robert Israel, <a href="/A308900/b308900.txt">Table of n, a(n) for n = 1..1999</a>
%H A308900 R. Israel, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2019-July/018885.html">Re: Help with a(n) and a cumulative sum</a>, Seqfan (Jul 15 2019).
%H A308900 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-1,10,10).
%F A308900 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).
%F A308900 From _Robert Israel_, Jul 15 2019: (Start)
%F A308900 G.f. (1+7*x)/((1+x)*(1-10*x^2)).
%F A308900 a(n) = -a(n - 1) + 10*a(n - 2) + 10*a(n - 3). (End)
%F A308900 a(-n) = a(n+1). - _Paul Curtz_, Jul 18 2019
%F A308900 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
%p A308900 1, seq(op([6*(10^i-1)/9, 3*(10^i-1)/9+1]), i=1..30); # _Robert Israel_, Jul 15 2019
%t A308900 CoefficientList[Series[(1 + 7 x)/((1 + x) (1 - 10 x^2)), {x, 0, 26}], x] (* _Michael De Vlieger_, Jul 18 2019 *)
%t A308900 LinearRecurrence[{-1,10,10},{1,6,4},30] (* _Harvey P. Dale_, Jan 02 2022 *)
%o A308900 (PARI) Vec((1+7*x)/((1+x)*(1-10*x^2)) + O(x^20)) \\ _Felix Fröhlich_, Jul 15 2019
%o A308900 (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
%o A308900 (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
%Y A308900 Cf. A002280, A093137, A309151.
%Y A308900 Cf. A289404.
%K A308900 nonn,base,easy
%O A308900 1,2
%A A308900 _N. J. A. Sloane_, Jul 15 2019