|
%I #48 Mar 29 2024 08:44:51
%S 1,1,4,6,14,26,54,106,214,426,854,1706,3414,6826,13654,27306,54614,
%T 109226,218454,436906,873814,1747626,3495254,6990506,13981014,
%U 27962026,55924054,111848106,223696214,447392426,894784854,1789569706,3579139414
%N Inverse binomial transform of a math magic problem.
%C Inverse binomial transform of A060816.
%H Vincenzo Librandi, <a href="/A084214/b084214.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,2).
%F a(n) = (5*2^n - 3*0^n + 4*(-1)^n)/6.
%F G.f.: (1+x^2)/((1+x)*(1-2*x)).
%F E.g.f.: (5*exp(2*x) - 3*exp(0) + 4*exp(-x))/6.
%F From _Paul Barry_, May 04 2004: (Start)
%F The binomial transform of a(n+1) is A020989(n).
%F a(n) = A001045(n-1) + A001045(n+1) - 0^n/2. (End)
%F a(n) = Sum_{k=0..n} A001045(n+1)*C(1, k/2)*(1+(-1)^k)/2. - _Paul Barry_, Oct 15 2004
%F a(n) = a(n-1) + 2*a(n-2) for n > 2. - _Klaus Brockhaus_, Dec 01 2009
%F From _Yuchun Ji_, Mar 18 2019: (Start)
%F a(n+1) = Sum_{i=0..n} a(i) + 1 - (-1)^n, a(0)=1.
%F a(n) = A000975(n-3)*10 + 5 + (-1)^(n-3), a(0)=1, a(1)=1, a(2)=4. (End)
%F a(n) = A081254(n) + (n-1 mod 2). - _Kevin Ryde_, Dec 20 2023
%p a[0]:=1:a[1]:=4:for n from 2 to 50 do a[n]:=a[n-1]+2*a[n-2]od: seq(a[n], n=-1..31); # _Zerinvary Lajos_, Dec 15 2008
%t f[n_]:=2/(n+1);x=3;Table[x=f[x];Numerator[x],{n,0,5!}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 12 2010 *)
%t LinearRecurrence[{1,2},{1,1,4},50] (* _Harvey P. Dale_, Mar 05 2021 *)
%o (Magma) [(5*2^n-3*0^n+4*(-1)^n)/6: n in [0..35]]; // _Vincenzo Librandi_, Jun 15 2011
%o (Haskell)
%o a084214 n = a084214_list !! n
%o a084214_list = 1 : xs where
%o xs = 1 : 4 : zipWith (+) (map (* 2) xs) (tail xs)
%o -- _Reinhard Zumkeller_, Aug 01 2011
%o (PARI) a(n) = 5<<(n-1)\3 + bitnegimply(1,n); \\ _Kevin Ryde_, Dec 20 2023
%Y Cf. A000975, A001045, A020989, A048654, A060816, A081254.
%K easy,nonn
%O 0,3
%A _Paul Barry_, May 19 2003
| 