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a(n) = 1 + Fibonacci(n) - (Fibonacci(n) mod 2).
1

%I #15 Jul 08 2022 08:23:34

%S 1,1,1,3,3,5,9,13,21,35,55,89,145,233,377,611,987,1597,2585,4181,6765,

%T 10947,17711,28657,46369,75025,121393,196419,317811,514229,832041,

%U 1346269,2178309,3524579,5702887,9227465,14930353,24157817,39088169

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

%C All-odd Fibonacci sequence.

%H G. C. Greubel, <a href="/A104220/b104220.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,-1,-1).

%F G.f.: (1-x^2-x^4)/((1-x)*(1-x-x^2)*(1+x+x^2)). [From Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009]

%F a(n) = (1 + 3*Fibonacci(n) + 2*(-1)^n*ChebyshevT(n, 1/2))/3. - _G. C. Greubel_, Jul 07 2022

%t Table[1+Fibonacci[n] -Mod[Fibonacci[n], 2], {n,0,60}]

%o (Magma) [Fibonacci(n) -(Fibonacci(n) mod 2) +1: n in [0..50]]; // _G. C. Greubel_, Jul 07 2022

%o (SageMath) [fibonacci(n) - (fibonacci(n)%2) + 1 for n in (0..50)] # _G. C. Greubel_, Jul 07 2022

%Y Cf. A000045.

%K nonn

%O 0,4

%A _Roger L. Bagula_, Mar 14 2005