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%I #16 Sep 08 2022 08:45:13
%S 0,1,1,3,3,8,8,21,21,55,55,144,144,377,377,987,987,2584,2584,6765,
%T 6765,17711,17711,46368,46368,121393,121393,317811,317811,832040,
%U 832040,2178309,2178309,5702887,5702887,14930352,14930352,39088169,39088169
%N Left-hand neighbors of Fibonacci numbers in Stern's diatomic series.
%C Fibonacci(2n) repeated. a(n) is the left neighbor of Fibonacci(n+2) in A002487 and A049456. A000045(n+2) = a(n)+A094967(n).
%H Colin Barker, <a href="/A094966/b094966.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 (0,3,0,-1).
%F G.f.: x*(1+x) / (1-3*x^2+x^4).
%F a(n) = Fibonacci(n)*(1+(-1)^n)/2 + Fibonacci(n+1)*(1-(-1)^n)/2.
%F a(n) = (2^(-2-n)*((1-sqrt(5))^n*(-3+sqrt(5)) - (-1-sqrt(5))^n*(-1+sqrt(5)) - (-1+sqrt(5))^n - sqrt(5)*(-1+sqrt(5))^n + 3*(1+sqrt(5))^n + sqrt(5)*(1+sqrt(5))^n))/sqrt(5). - _Colin Barker_, Mar 28 2016
%t CoefficientList[Series[x (1 + x)/(1 - 3 x^2 + x^4), {x, 0, 38}], x] (* _Michael De Vlieger_, Mar 28 2016 *)
%o (PARI) concat(0, Vec(x*(1+x)/(1-3*x^2+x^4) + O(x^50))) \\ _Colin Barker_, Mar 28 2016
%o (Magma) [Fibonacci(n)*(1+(-1)^n)/2 + Fibonacci(n+1)*(1-(-1)^n)/2: n in [0..40]]; // _Vincenzo Librandi_, Mar 29 2016
%Y Cf. A001906.
%K easy,less,nonn
%O 0,4
%A _Paul Barry_, May 26 2004
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