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

Offset

0,3

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).

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

Maple

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

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 *)

Prog

(Magma) [(5*2^n-3*0^n+4*(-1)^n)/6: n in [0..35]]; // Vincenzo Librandi, Jun 15 2011
(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
(PARI) a(n) = 5<<(n-1)\3 + bitnegimply(1, n); \\ Kevin Ryde, Dec 20 2023

Crossrefs

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

Sequence in context: A192782 A306742 A188576 * A030138 A369684 A009849

Adjacent sequences: A084211 A084212 A084213 * A084215 A084216 A084217

Keyword

easy,nonn

Author

Paul Barry, May 19 2003

Status

approved