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

1, 1, 2, 2, 5, 5, 13, 13, 34, 34, 89, 89, 233, 233, 610, 610, 1597, 1597, 4181, 4181, 10946, 10946, 28657, 28657, 75025, 75025, 196418, 196418, 514229, 514229, 1346269, 1346269, 3524578, 3524578, 9227465, 9227465, 24157817, 24157817, 63245986, 63245986, 165580141, 165580141

Offset: 0

Author

Paul Barry, May 26 2004

Keywords

• easy
• nonn

Comments

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).

Diagonal sums of A109223. - Paul Barry, Jun 22 2005

The Fi2 sums, see A180662, of triangle A065941 equal the terms of this sequence. - Johannes W. Meijer, Aug 11 2011

a(n) is the last term of (n+1)-th row in Wythoff array A003603. -Reinhard Zumkeller, Jan 26 2012

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..2000
• Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 15.
• Ying Wang and Zihao Zhang, Hankel Determinants for a Class of Weighted Lattice Paths, arXiv:2409.18609 [math.CO], 2024. See p. 2.
• Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1).

Crossrefs

Cf. A001519, A133080.

Programs

GAP

List([0..50], n -> Fibonacci(n)*(1-(-1)^n)/2 + Fibonacci(n+1)*(1+(-1)^n)/2); # G. C. Greubel, Nov 18 2018

Magma

[IsEven(n) select Fibonacci(n+1) else Fibonacci(n): n in [0..70]]; // Vincenzo Librandi, Nov 18 2018

Maple

A094967 := proc(n) combinat[fibonacci](2*floor(n/2)+1) ; end proc: seq(A094967(n), n=0..37);

Mathematica

LinearRecurrence[{0,3,0,-1},{1,1,2,2},40] (* Harvey P. Dale, Apr 05 2015 *)
f[n_]:=If[OddQ@n, (Fibonacci[n]), Fibonacci[n+1]]; Array[f, 100, 0] (* Vincenzo Librandi, Nov 18 2018 *)
Table[Fibonacci[n, 0]*Fibonacci[n] + LucasL[n, 0]*Fibonacci[n + 1]/2, {n, 0, 50}] (* G. C. Greubel, Nov 18 2018 *)

PARI

vector(50, n, n--; fibonacci(n)*(1-(-1)^n)/2 + fibonacci(n+1)*(1+(-1)^n)/2) \\ G. C. Greubel, Nov 18 2018

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

Formula

G.f.: (1+x-x^2-x^3)/(1-3*x^2+x^4).

a(n) = Fibonacci(n)*(1-(-1)^n)/2 + Fibonacci(n+1)*(1+(-1)^n)/2.

a(n) = Sum_{k=0..floor(n/2)} binomial(floor(n/2)+k, 2*k). - Paul Barry, Jun 22 2005

Starting (1, 2, 2, 5, 5, 13, 13, ...) = A133080 * A000045, where A000045 starts with "1". - Gary W. Adamson, Sep 08 2007

a(n) = Fibonacci(n+1)^(4*k+3) mod Fibonacci(n+2), for any k>-1, n>0. - Gary Detlefs, Nov 29 2010