A014113 a(n) = a(n-1) + 2*a(n-2) with a(0)=0, a(1)=2.

0, 2, 2, 6, 10, 22, 42, 86, 170, 342, 682, 1366, 2730, 5462, 10922, 21846, 43690, 87382, 174762, 349526, 699050, 1398102, 2796202, 5592406, 11184810, 22369622, 44739242, 89478486, 178956970, 357913942, 715827882, 1431655766, 2863311530, 5726623062, 11453246122

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 457

Index entries for linear recurrences with constant coefficients, signature (1,2).

Formula

a(0) = 0 and if n>=1, a(n) = 2^n - a(n-1).

a(n) = A078008(n+1). - Reinhard Zumkeller, Jun 10 2005

a(n) = 2*A001045(n). - Benoit Jubin, Dec 01 2011

a(n) = (2^(n+1) - 2*(-1)^n)/3. - Ralf Stephan, Jul 18 2013

G.f.: 2*x/(1+x)/(1-2*x). - Colin Barker, Feb 19 2012

G.f.: 1/Q(0) -1, where Q(k) = 1 + 2*x^2 - (2*k+3)*x + x*(2*k+1 - 2*x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 05 2013

Consider a Pascal-like triangle T(n,k) in which T(n,0) = T(n,n) = 0 if n even, 1 if n odd, and each interior entry T(n,k) is the sum of the two entries "above" it: T(n,k) = T(n-1,k-1) + T(n-1,k) for 0 < k < n. Then, a(n) is the sum of the entries in the n-th row of T(n,k). - Greg Dresden, May 24 2024

Mathematica

LinearRecurrence[{1, 2}, {0, 2}, 50] (* Vincenzo Librandi, Feb 19 2012 *)

Prog

(Haskell)
a014113 n = a014113_list !! n
a014113_list = 0 : 2 : zipWith (+)
                       (map (* 2) a014113_list) (tail a014113_list)
-- Reinhard Zumkeller, Jul 20 2013

Crossrefs

Cf. A001045, A078008, A163271, A290732.

Sequence in context: A019310 A078008 A151575 * A284462 A262278 A265639

Adjacent sequences: A014110 A014111 A014112 * A014114 A014115 A014116

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved