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A094967 Right-hand neighbors of Fibonacci numbers in Stern's diatomic series. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1).

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

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 *)

PROG

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

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

CROSSREFS

Cf. A001519, A133080.

Sequence in context: A056504 A122205 A178115 * A322111 A321395 A056505

Adjacent sequences:  A094964 A094965 A094966 * A094968 A094969 A094970

KEYWORD

easy,nonn

AUTHOR

Paul Barry, May 26 2004

STATUS

approved

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Last modified January 21 11:00 EST 2019. Contains 319351 sequences. (Running on oeis4.)