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A083313 a(0)=1; a(n) = 3^n - 2^(n-1) for n >= 1.

%I A083313 #48 Sep 08 2022 08:45:10
%S A083313 1,2,7,23,73,227,697,2123,6433,19427,58537,176123,529393,1590227,
%T A083313 4774777,14332523,43013953,129074627,387289417,1161999323,3486260113,
%U A083313 10459304627,31378962457,94138984523,282421147873,847271832227,2541832273897,7625530376123
%N A083313 a(0)=1; a(n) = 3^n - 2^(n-1) for n >= 1.
%C A083313 Essentially the same as A064686.
%C A083313 Binomial transform of A051049.
%C A083313 Number of skinny Boolean functions f(x_1,...,x_n) that are also Horn functions. - _Hugo Pfoertner_, Mar 04 2019
%D A083313 Donald E. Knuth, The Art of Computer Programming, Vol. 4, fascicle 1, section 7.1.4, pp. 134, 138, 139, 219, answer to exercise 172, Addison-Wesley, 2009.
%H A083313 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-6).
%F A083313 a(n) = (2*3^n - (2^n - 0^n))/2.
%F A083313 a(0) = 1, a(n) = 3^n - 2^(n-1) for n >= 1.
%F A083313 G.f.: ((1-x) + (1-2*x)*(1-3*x))/(2*(1-2*x)*(1-3*x)).
%F A083313 E.g.f.: (2*exp(3*x) - exp(2*x) + exp(0))/2.
%F A083313 a(n) = A090888(n-1, 4), for n > 0. - _Ross La Haye_, Sep 21 2004
%F A083313 Let b(n) = 2*(3/2)^n - 1. Then A003063(n) = -b(1-n)*3^(n-1) for n > 0. a(n) = A064686(n) = b(n)*2^(n-1) for n > 0. - _Michael Somos_, Aug 06 2006
%F A083313 From _Alex Ratushnyak_, Jul 03 2012: (Start)
%F A083313 a(n) mod 100 = 23 for n = 4*k-1, k >= 1.
%F A083313 a(n) mod 100 = 27 for n = 4*k+1, k >= 1.
%F A083313 (End)
%p A083313 A083313 := proc(n)
%p A083313     if n = 0 then
%p A083313         1;
%p A083313     else
%p A083313         3^n-2^(n-1) ;
%p A083313     end if;
%p A083313 end proc: # _R. J. Mathar_, Aug 01 2013
%t A083313 CoefficientList[Series[((1 - x) + (1 - 2 x) (1 - 3 x)) / (2 (1 - 2 x) (1 - 3 x)), {x, 0, 30}], x] (* _Vincenzo Librandi_, Feb 01 2015 *)
%t A083313 LinearRecurrence[{5,-6},{1,2,7},30] (* _Harvey P. Dale_, Sep 04 2017 *)
%o A083313 (PARI) Vec(((1-x)+(1-2*x)*(1-3*x))/(2*(1-2*x)*(1-3*x)) + O(x^30)) \\ _Michel Marcus_, Jan 31 2015
%o A083313 (PARI)  print1(1,", ",s=2,", " );for(k=2,27,s=2^(k-2)+3*s;print1(s,", ")) \\ _Hugo Pfoertner_, Mar 04 2019
%o A083313 (Magma) [(2*3^n-2^n+0^n)/2: n in [0..30]]; // _Vincenzo Librandi_, Feb 01 2015
%Y A083313 Cf. A083314.
%K A083313 easy,nonn
%O A083313 0,2
%A A083313 _Paul Barry_, Apr 24 2003
%E A083313 Better name by _Alex Ratushnyak_, Jul 02 2012