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%I #10 Jan 01 2024 11:43:17
%S 0,1,1,3,7,17,39,87,189,403,847,1761,3631,7439,15165,30795,62343,
%T 125905,253783,510759,1026685,2061731,4136991,8295873,16627167,
%U 33311647,66716029,133582107,267406999,535206833,1071049287
%N 2^n + 1 - 2*Fibonacci(n+1).
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4,-1,2).
%F a(n) = 2^n + 1 - 2*Fibonacci(n+1) = 2^n + 1 + Fibonacci(n) - Fibonacci(n+3) = 2^n + 1 - Fibonacci(n) - Lucas(n). a(n) = 2(2^(n-1) - Fibonacci(n+1)) + 1, for n > 0. a(n) = A000051(n) - A006355(n+2) = A000051(n) - A000045(n) - A000032(n). a(n) = A101220(2,2,n-1) - A101220(1,1,n-3), for n > 2. a(n) = A008466(n) - A000071(n-1), for n > 0. a(n) = 2*A008466(n-1) + 1, for n > 0.
%F a(n) = 2*A101220(2,2,n-2) + 1, for n > 1. a(n) = Sum[2^(n-k)Fibonacci(k) - Fibonacci(k-2),{k,0,n}] = antidiagonal sums of A118654. a(n+1) - a(n) = 2(2^(n-1) - Fibonacci(n)), for n > 0. a(n+1) - a(n) = 2*A027934(n-2), for n > 1. a(n+1) - a(n) = 2*A101220(1,2,n-1), for n > 0. a(0) = 0; a(1) = 1; a(n) = a(n-1) + a(n-2) + 2^(n-2) - 1, for n > 1. a(0) = 0; a(1) = 1; a(2) = 1; a(3) = 3; a(n) = 4*a(n-1) - 4*a(n-2) - a(n-3) + 2*a(n-4), for n > 3.
%F O.g.f. = x(1-3x+3x^2)/((1-x)(1-2x)(1-x-x^2)).
%t Table[2^n + 1 - 2 Fibonacci[n + 1], {n, 0, 30}]
%Y Cf. A000032, A000045, A000051, A000071, A006355, A008466, A027934, A101220, A118654.
%K nonn
%O 0,4
%A _Ross La Haye_, May 31 2006, Jun 27 2007