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A094967 Right-hand neighbors of Fibonacci numbers in Stern's diatomic series.

%I A094967 #45 Oct 08 2024 18:39:05
%S A094967 1,1,2,2,5,5,13,13,34,34,89,89,233,233,610,610,1597,1597,4181,4181,
%T A094967 10946,10946,28657,28657,75025,75025,196418,196418,514229,514229,
%U A094967 1346269,1346269,3524578,3524578,9227465,9227465,24157817,24157817,63245986,63245986,165580141,165580141
%N A094967 Right-hand neighbors of Fibonacci numbers in Stern's diatomic series.
%C A094967 Fibonacci(2*n+1) repeated. a(n) is the right neighbor of Fibonacci(n+2) in A049456 and A002487 (starts 2,2,5,...). A000045(n+2) = A094966(n) + a(n).
%C A094967 Diagonal sums of A109223. - _Paul Barry_, Jun 22 2005
%C A094967 The Fi2 sums, see A180662, of triangle A065941 equal the terms of this sequence. - _Johannes W. Meijer_, Aug 11 2011
%C A094967 a(n) is the last term of (n+1)-th row in Wythoff array A003603. -_Reinhard Zumkeller_, Jan 26 2012
%H A094967 Vincenzo Librandi, <a href="/A094967/b094967.txt">Table of n, a(n) for n = 0..2000</a>
%H A094967 Jia Huang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Huang/huang8.html">Partially Palindromic Compositions</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 15.
%H A094967 Ying Wang and Zihao Zhang, <a href="https://arxiv.org/abs/2409.18609">Hankel Determinants for a Class of Weighted Lattice Paths</a>, arXiv:2409.18609 [math.CO], 2024. See p. 2.
%H A094967 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-1).
%F A094967 G.f.: (1+x-x^2-x^3)/(1-3*x^2+x^4).
%F A094967 a(n) = Fibonacci(n)*(1-(-1)^n)/2 + Fibonacci(n+1)*(1+(-1)^n)/2.
%F A094967 a(n) = Sum_{k=0..floor(n/2)} binomial(floor(n/2)+k, 2*k). - _Paul Barry_, Jun 22 2005
%F A094967 Starting (1, 2, 2, 5, 5, 13, 13, ...) = A133080 * A000045, where A000045 starts with "1". - _Gary W. Adamson_, Sep 08 2007
%F A094967 a(n) = Fibonacci(n+1)^(4*k+3) mod Fibonacci(n+2), for any k>-1, n>0. - _Gary Detlefs_, Nov 29 2010
%p A094967 A094967 := proc(n) combinat[fibonacci](2*floor(n/2)+1) ; end proc: seq(A094967(n), n=0..37);
%t A094967 LinearRecurrence[{0,3,0,-1},{1,1,2,2},40] (* _Harvey P. Dale_, Apr 05 2015 *)
%t A094967 f[n_]:=If[OddQ@n, (Fibonacci[n]), Fibonacci[n+1]]; Array[f, 100, 0] (* _Vincenzo Librandi_, Nov 18 2018 *)
%t A094967 Table[Fibonacci[n, 0]*Fibonacci[n] + LucasL[n, 0]*Fibonacci[n + 1]/2, {n, 0, 50}] (* _G. C. Greubel_, Nov 18 2018 *)
%o A094967 (Magma) [IsEven(n) select Fibonacci(n+1) else Fibonacci(n): n in [0..70]]; // _Vincenzo Librandi_, Nov 18 2018
%o A094967 (PARI) vector(50, n, n--; fibonacci(n)*(1-(-1)^n)/2 + fibonacci(n+1)*(1+(-1)^n)/2) \\ _G. C. Greubel_, Nov 18 2018
%o A094967 (Sage) [fibonacci(n)*(1-(-1)^n)/2 + fibonacci(n+1)*(1+(-1)^n)/2 for n in range(50)] # _G. C. Greubel_, Nov 18 2018
%o A094967 (GAP) List([0..50], n -> Fibonacci(n)*(1-(-1)^n)/2 + Fibonacci(n+1)*(1+(-1)^n)/2); # _G. C. Greubel_, Nov 18 2018
%Y A094967 Cf. A001519, A133080.
%K A094967 easy,nonn
%O A094967 0,3
%A A094967 _Paul Barry_, May 26 2004