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A272263 a(n) = numerator of A000032(n) - 1/2^n.

%I A272263 #19 Feb 18 2019 14:08:08
%S A272263 1,1,11,31,111,351,1151,3711,12031,38911,125951,407551,1318911,
%T A272263 4268031,13811711,44695551,144637951,468058111,1514668031,4901568511,
%U A272263 15861809151,51329892351,166107021311,537533612031,1739495309311,5629125066751,18216231370751
%N A272263 a(n) = numerator of A000032(n) - 1/2^n.
%C A272263 A000032(n), Lucas numbers, and 1/2^n are autosequences of the second kind.
%C A272263 Then a(n)/2^n is also an autosequence of the second kind.
%C A272263 Their corresponding autosequences of the first kind are A000045(n) and n/2^n, the Oresme numbers.
%C A272263 Difference table of A000032(n) - 1/2^n:
%C A272263 1, 1/2,  11/4,   31/8,  111/16,    351/32,  1151/64, ...
%C A272263    9/4,   9/8,  49/16,  129/32,    449/64, 1409/128, ...
%C A272263         31/16,  31/32,  191/64,   511/128, 1791/256, ...
%C A272263                129/64, 129/128,   769/256, ...
%C A272263                        511/256,   511/256, ...
%C A272263                                 2049/1024, ... .
%C A272263 The first upper diagonal is A140323(n)/A004171(n). The main diagonal is the double, i.e. A140323(n)/A000302(n). The inverse binomial transform is the signed sequence.
%C A272263 Quintisections from a(2):
%C A272263      11,      31,      111,      351,      1151,
%C A272263    3711,   12031,    38911,   125951,    407551,
%C A272263 1318911, 4268031, 13811711, 44695551, 144637951,
%C A272263 etc.
%H A272263 Colin Barker, <a href="/A272263/b272263.txt">Table of n, a(n) for n = 0..1000</a>
%H A272263 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,2,-4).
%F A272263 a(n) = a(n-1) + 10*A085449(n), for n>0, a(0)=1.
%F A272263 a(n) = A087131(n) - 1.
%F A272263 From _Colin Barker_, Apr 24 2016: (Start)
%F A272263 a(n) = (-1+(1-sqrt(5))^n+(1+sqrt(5))^n).
%F A272263 a(n) = 3*a(n-1)+2*a(n-2)-4*a(n-3) for n>2.
%F A272263 G.f.: (1-2*x+6*x^2) / ((1-x)*(1-2*x-4*x^2)).
%F A272263 (End)
%e A272263 Numerators of a(0) =2-1=1, a(1)=1-1/2=1/2, a(2)=3-1/4=11/4, a(3)=4-1/8=31/8, ... .
%t A272263 CoefficientList[Series[(1 - 2*x + 6*x^2)/((1 - x)*(1 - 2*x - 4*x^2)), {x, 0, 30}], x] (* _Robert Price_, Apr 24 2016 *)
%t A272263 Table[Numerator[LucasL@ n - 1/2^n], {n, 0, 26}] (* _Michael De Vlieger_, Apr 24 2016 *)
%o A272263 (PARI) Vec((1-2*x+6*x^2)/((1-x)*(1-2*x-4*x^2)) + O(x^50)) \\ _Colin Barker_, Apr 24 2016
%Y A272263 Cf. A000012, A000032, A000045, A000079, A000302, A004171, A085449, A087131, A140323.
%K A272263 nonn,frac,easy
%O A272263 0,3
%A A272263 _Paul Curtz_, Apr 24 2016