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A084214 Inverse binomial transform of a math magic problem.

%I A084214 #55 May 20 2025 14:45:17
%S A084214 1,1,4,6,14,26,54,106,214,426,854,1706,3414,6826,13654,27306,54614,
%T A084214 109226,218454,436906,873814,1747626,3495254,6990506,13981014,
%U A084214 27962026,55924054,111848106,223696214,447392426,894784854,1789569706,3579139414,7158278826,14316557654
%N A084214 Inverse binomial transform of a math magic problem.
%C A084214 Inverse binomial transform of A060816.
%H A084214 Vincenzo Librandi, <a href="/A084214/b084214.txt">Table of n, a(n) for n = 0..1000</a>
%H A084214 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,2).
%F A084214 a(n) = (5*2^n - 3*0^n + 4*(-1)^n)/6.
%F A084214 G.f.: (1+x^2)/((1+x)*(1-2*x)).
%F A084214 E.g.f.: (5*exp(2*x) - 3*exp(0) + 4*exp(-x))/6.
%F A084214 From _Paul Barry_, May 04 2004: (Start)
%F A084214 The binomial transform of a(n+1) is A020989(n).
%F A084214 a(n) = A001045(n-1) + A001045(n+1) - 0^n/2. (End)
%F A084214 a(n) = Sum_{k=0..n} A001045(n+1)*C(1, k/2)*(1+(-1)^k)/2. - _Paul Barry_, Oct 15 2004
%F A084214 a(n) = a(n-1) + 2*a(n-2) for n > 2. - _Klaus Brockhaus_, Dec 01 2009
%F A084214 From _Yuchun Ji_, Mar 18 2019: (Start)
%F A084214 a(n+1) = Sum_{i=0..n} a(i) + 1 - (-1)^n, a(0)=1.
%F A084214 a(n) = A000975(n-3)*10 + 5 + (-1)^(n-3), a(0)=1, a(1)=1, a(2)=4. (End)
%F A084214 a(n) = A081254(n) + (n-1 mod 2). - _Kevin Ryde_, Dec 20 2023
%F A084214 a(n) = 2*A048573(n-2) for n>=2. - _Alois P. Heinz_, May 20 2025
%p A084214 A084214 := proc(n)
%p A084214     (5*2^n - 3*0^n + 4*(-1)^n)/6 ;
%p A084214 end proc:
%p A084214 seq(A084214(n),n=0..60) ; # _R. J. Mathar_, Aug 18 2024
%t A084214 f[n_]:=2/(n+1);x=3;Table[x=f[x];Numerator[x],{n,0,5!}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 12 2010 *)
%t A084214 LinearRecurrence[{1,2},{1,1,4},50] (* _Harvey P. Dale_, Mar 05 2021 *)
%o A084214 (Magma) [(5*2^n-3*0^n+4*(-1)^n)/6: n in [0..35]]; // _Vincenzo Librandi_, Jun 15 2011
%o A084214 (Haskell)
%o A084214 a084214 n = a084214_list !! n
%o A084214 a084214_list = 1 : xs where
%o A084214    xs = 1 : 4 : zipWith (+) (map (* 2) xs) (tail xs)
%o A084214 -- _Reinhard Zumkeller_, Aug 01 2011
%o A084214 (PARI) a(n) = 5<<(n-1)\3 + bitnegimply(1,n); \\ _Kevin Ryde_, Dec 20 2023
%Y A084214 Cf. A000975, A001045, A020989, A048654, A060816, A081254.
%Y A084214 Cf. A048573.
%K A084214 easy,nonn
%O A084214 0,3
%A A084214 _Paul Barry_, May 19 2003