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A093142 Expansion of g.f. (1-5*x)/((1-x)*(1-10*x)).

%I A093142 #25 Apr 30 2025 09:16:30
%S A093142 1,6,56,556,5556,55556,555556,5555556,55555556,555555556,5555555556,
%T A093142 55555555556,555555555556,5555555555556,55555555555556,
%U A093142 555555555555556,5555555555555556,55555555555555556,555555555555555556,5555555555555555556,55555555555555555556,555555555555555555556
%N A093142 Expansion of g.f. (1-5*x)/((1-x)*(1-10*x)).
%C A093142 Second binomial transform of 5*A001045(3n)/3+(-1)^n.
%C A093142 Partial sums of A093143.
%C A093142 A convex combination of 10^n and 1.
%C A093142 In general the second binomial transform of k*Jacobsthal(3n)/3+(-1)^n is 1, 1+k, 1+11k, 1+111k, ... This is the case for k=5.
%H A093142 Seiichi Manyama, <a href="/A093142/b093142.txt">Table of n, a(n) for n = 0..1000</a>
%H A093142 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (11,-10).
%F A093142 a(n) = 5*10^n/9 + 4/9.
%F A093142 a(n) = 10*a(n-1) - 4 with a(0)=1. - _Vincenzo Librandi_, Aug 02 2010
%F A093142 a(n) = 11*a(n-1) - 10*a(n-2), n > 1. - _Harvey P. Dale_, Aug 23 2014
%F A093142 From _Elmo R. Oliveira_, Apr 29 2025: (Start)
%F A093142 E.g.f.: exp(x)*(5*exp(9*x) + 4)/9.
%F A093142 a(n) = (A062397(n) + A002275(n))/2. (End)
%t A093142 CoefficientList[Series[(1-5x)/((1-x)(1-10x)),{x,0,20}],x] (* or *) LinearRecurrence[{11,-10},{1,6},20] (* _Harvey P. Dale_, Aug 23 2014 *)
%o A093142 (PARI) {a(n) = (5*10^n+4)/9} \\ _Seiichi Manyama_, Sep 14 2019
%Y A093142 Cf. A001045, A002275, A062397, A093143.
%K A093142 easy,nonn
%O A093142 0,2
%A A093142 _Paul Barry_, Mar 24 2004