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
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).
Programs
-
Haskell
a014113 n = a014113_list !! n a014113_list = 0 : 2 : zipWith (+) (map (* 2) a014113_list) (tail a014113_list) -- Reinhard Zumkeller, Jul 20 2013 -
Mathematica
LinearRecurrence[{1,2},{0,2},50] (* Vincenzo Librandi, Feb 19 2012 *)
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