A084214 Inverse binomial transform of a math magic problem.

1, 1, 4, 6, 14, 26, 54, 106, 214, 426, 854, 1706, 3414, 6826, 13654, 27306, 54614, 109226, 218454, 436906, 873814, 1747626, 3495254, 6990506, 13981014, 27962026, 55924054, 111848106, 223696214, 447392426, 894784854, 1789569706, 3579139414, 7158278826, 14316557654

Offset: 0

Author

Paul Barry, May 19 2003

Keywords

• easy
• nonn

Comments

Inverse binomial transform of A060816.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,2).

Crossrefs

Cf. A000975, A001045, A020989, A048654, A060816, A081254.

Cf. A048573.

Programs

Haskell

a084214 n = a084214_list !! n
a084214_list = 1 : xs where
   xs = 1 : 4 : zipWith (+) (map (* 2) xs) (tail xs)
-- Reinhard Zumkeller, Aug 01 2011

Magma

[(5*2^n-3*0^n+4*(-1)^n)/6: n in [0..35]]; // Vincenzo Librandi, Jun 15 2011

Maple

A084214 := proc(n)
    (5*2^n - 3*0^n + 4*(-1)^n)/6 ;
end proc:
seq(A084214(n),n=0..60) ; # R. J. Mathar, Aug 18 2024

Mathematica

f[n_]:=2/(n+1);x=3;Table[x=f[x];Numerator[x],{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Mar 12 2010 *)
LinearRecurrence[{1,2},{1,1,4},50] (* Harvey P. Dale, Mar 05 2021 *)

PARI

a(n) = 5<<(n-1)\3 + bitnegimply(1,n); \\ Kevin Ryde, Dec 20 2023

Formula

a(n) = (5*2^n - 3*0^n + 4*(-1)^n)/6.

G.f.: (1+x^2)/((1+x)*(1-2*x)).

E.g.f.: (5*exp(2*x) - 3*exp(0) + 4*exp(-x))/6.

From Paul Barry, May 04 2004: (Start)

The binomial transform of a(n+1) is A020989(n).

a(n) = A001045(n-1) + A001045(n+1) - 0^n/2. (End)

a(n) = Sum_{k=0..n} A001045(n+1)*C(1, k/2)*(1+(-1)^k)/2. - Paul Barry, Oct 15 2004

a(n) = a(n-1) + 2*a(n-2) for n > 2. - Klaus Brockhaus, Dec 01 2009

From Yuchun Ji, Mar 18 2019: (Start)

a(n+1) = Sum_{i=0..n} a(i) + 1 - (-1)^n, a(0)=1.

a(n) = A000975(n-3)*10 + 5 + (-1)^(n-3), a(0)=1, a(1)=1, a(2)=4. (End)

a(n) = A081254(n) + (n-1 mod 2). - Kevin Ryde, Dec 20 2023

a(n) = 2*A048573(n-2) for n>=2. - Alois P. Heinz, May 20 2025