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a(n) = 2^n - Fibonacci(n) - 1.
2

%I #19 Sep 08 2022 08:46:05

%S 0,0,2,5,12,26,55,114,234,477,968,1958,3951,7958,16006,32157,64548,

%T 129474,259559,520106,1041810,2086205,4176592,8359950,16730847,

%U 33479406,66987470,134021309,268117644,536356682,1072909783,2146137378,4292788986,8586410013

%N a(n) = 2^n - Fibonacci(n) - 1.

%C a(n+1) = sum of n-th row of the triangle in A228074.

%H Reinhard Zumkeller, <a href="/A228078/b228078.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4,-1,2).

%F a(n) = A000079(n) - A000045(n) - 1 = A000225(n) - A000045(n) = A000079(n) - A001611(n) = A099036(n) - 1.

%F a(n) = 4*a(n-1)-4*a(n-2)-a(n-3)+2*a(n-4) for n>3. - _Colin Barker_, Mar 20 2015

%F G.f.: x^2*(3*x-2) / ((x-1)*(2*x-1)*(x^2+x-1)). - _Colin Barker_, Mar 20 2015

%F a(n) = (-1+2^n+(((1-sqrt(5))/2)^n-((1+sqrt(5))/2)^n)/sqrt(5)). - _Colin Barker_, Nov 02 2016

%t Table[(2^n - Fibonacci[n] - 1), {n, 0, 40}] (* _Vincenzo Librandi_, Aug 16 2013 *)

%o (Haskell)

%o a228078 = subtract 1 . a099036

%o (Magma)

%o [2^n - Fibonacci(n) - 1: n in [0..40]]; // _Vincenzo Librandi_, Aug 16 2013

%o (PARI) concat([0,0], Vec(x^2*(3*x-2)/((x-1)*(2*x-1)*(x^2+x-1)) + O(x^100))) \\ _Colin Barker_, Mar 20 2015

%K nonn,easy

%O 0,3

%A _Reinhard Zumkeller_, Aug 15 2013