A094966 -id:A094966 - cp's OEIS frontend

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

Paul Barry, May 26 2004

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

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).

Cf. A001519, A133080.

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

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

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

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 *)

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

[fibonacci(n)*(1-(-1)^n)/2 + fibonacci(n+1)*(1+(-1)^n)/2 for n in range(50)] # G. C. Greubel, Nov 18 2018

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

A056014 a(n) = (Fibonacci(2n-1) - Fibonacci(n+1))/2.

Original entry on oeis.org

0, 0, 0, 1, 4, 13, 38, 106, 288, 771, 2046, 5401, 14212, 37324, 97904, 256621, 672336, 1760997, 4611642, 12075526, 31617520, 82781215, 216732890, 567428401, 1485570024, 3889310328, 10182407328, 26657986681, 69791674108

Offset: 0

Asher Auel, Jun 06 2000

With a(0)=0, a(1)=1, a(2)=1, a(3)=2, this recurrence produces a(n)=A000045(n) (Fibonacci numbers).

Number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 5 and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 1, s(n) = 4. - Herbert Kociemba, Jun 16 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Éva Czabarka, Rigoberto Flórez, and Leandro Junes, A Discrete Convolution on the Generalized Hosoya Triangle, Journal of Integer Sequences, 18 (2015), Article 15.1.6.
Henrik Eriksson and Markus Jonsson, Level sizes of the Bulgarian solitaire game tree, Fib. Q., 35:3 (2017), 243-251.
Hideyuki Othsuka, Problem B-1345, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 62, No. 1 (2024), p. 85.
Index entries for linear recurrences with constant coefficients, signature (4,-3,-2,1).

Cf. A000045, A000079, A051450, A056015, A094966.

a(1-2n)=A059512(2n), a(-2n)=A027994(2n-1).

I:=[0, 0, 0, 1]; [n le 4 select I[n] else 4*Self(n-1)-3*Self(n-2)-2*Self(n-3)+Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jun 23 2012

Table[(Fibonacci[2n-1]-Fibonacci[n+1])/2,{n,0,40}]  (* Harvey P. Dale, Mar 24 2011 *)
LinearRecurrence[{4,-3,-2,1},{0,0, 0,1},40] (* Vincenzo Librandi, Jun 23 2012 *)

a(n)=(fibonacci(2*n-1)-fibonacci(n+1))/2

a(n) = 4*a(n-1) - 3*a(n-2) - 2*a(n-3) + a(n-4), a(0)=a(1)=a(2)=0, a(3)=1.
Convolution of Fibonacci numbers F(n) with F(2n). - Benoit Cloitre, Jun 07 2004
G.f.: x^3/((1 - x - x^2)*(1 - 3*x + x^2)). - Herbert Kociemba, Jun 16 2004
Binomial transform of x^3/(1-3x^2+x^4), or (essentially) F(2n) with interpolated zeros. a(n)=sum{k=0..n, binomial(n, k)((3/2-sqrt(5)/2)^(k/2)((sqrt(5)/20+1/4)(-1)^k-sqrt(5)/20-1/4)+ (sqrt(5)/2+3/2)^(k/2)((sqrt(5)/20-1/4)(-1)^k-sqrt(5)/20+1/4))}. - Paul Barry, Jul 26 2004
Convolution of the powers of 2 (A000079) with the number of positive rational knots with 2n+1 crossings (A051450), with three leading zeros. - Graeme McRae, Jun 28 2006
a(n) = (A001519(n) - A000045(n+1))/2. - R. J. Mathar, Jun 24 2011
a(n) = Sum_{k=1..n-1} binomial(n-1, k) * A094966(k-1) (Othsuka, 2024). - Amiram Eldar, Feb 29 2024

A124377 Riordan array (1/(1-x-x^2),x/(1+x)).

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 3, 2, -1, 1, 5, 1, 3, -2, 1, 8, 4, -2, 5, -3, 1, 13, 4, 6, -7, 8, -4, 1, 21, 9, -2, 13, -15, 12, -5, 1, 34, 12, 11, -15, 28, -27, 17, -6, 1, 55, 22, 1, 26, -43, 55, -44, 23, -7, 1, 89, 33, 21

Offset: 0

Paul Barry, Oct 29 2006

First column is F(n+1). Second column is A008346. Row sums are F(n+2). Diagonal sums are A094966(n+1). Product of A007318 and A124377 is the Riordan array ((1-x)/(1-3x+x^2),x), the sequence array for F(2n+1).

			Triangle begins
1,
1, 1,
2, 0, 1,
3, 2, -1, 1,
5, 1, 3, -2, 1,
8, 4, -2, 5, -3, 1,
13, 4, 6, -7, 8, -4, 1,
21, 9, -2, 13, -15, 12, -5, 1

Cf. A000045

Number triangle T(n,k)=sum{j=0..n-k, C(j-k,n-k-j)}*[k<=n]
T(n,k)=T(n-1,k-1)+2*T(n-2,k)-T(n-2,k-1)+T(n-3,k)-T(n-3,k-1), T(0,0)=T(1,0)=T(1,1)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Jan 12 2014
T(n,0)=A000045(n+1), T(n,n)=1, T(n,k)=T(n-1,k-1)-T(n-1,k) for 0Philippe Deléham, Jan 12 2014

A140833 Sum of Fibonacci numbers between F(-n)....F(n), inclusive.

Original entry on oeis.org

0, 2, 2, 6, 6, 16, 16, 42, 42, 110, 110, 288, 288, 754, 754, 1974, 1974, 5168, 5168, 13530, 13530, 35422, 35422, 92736, 92736, 242786, 242786, 635622, 635622, 1664080, 1664080, 4356618, 4356618, 11405774, 11405774, 29860704, 29860704, 78176338, 78176338

Offset: 0

Carey W. Strutz (cwstrutz(AT)excite.com), Jul 18 2008

a(2n)/a(2n+1) converges to ((((sqrt 5)-1)/2)^2).

			a(3) = 2+(-1)+1+0+1+1+2=6.
G.f. = 2*x + 2*x^2 + 6*x^3 + 6*x^4 + 16*x^5 + 16*x^6 + 42*x^7 + ...

Matthew House, Table of n, a(n) for n = 0..4760
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1).

Cf. A000045, A025169, A001906.

a:= n-> 2*(<<0|1>, <1|1>>^(ceil(n/2)*2))[1,2]:
seq(a(n), n=0..40);  # Alois P. Heinz, Nov 02 2016

a[ n_] := 2 Fibonacci[ n + Mod[n, 2]]; (* Michael Somos, Nov 01 2016 *)
LinearRecurrence[{0,3,0,-1},{0,2,2,6},50] (* Harvey P. Dale, Aug 07 2022 *)

{a(n) = 2 * fibonacci(n + n%2)}; /* Michael Somos, Nov 01 2016 */

a(2n-1) = a(2n).
a(n) = 3*a(n-2) - a(n-4).
G.f.: 2x(1+x)/((1-x-x^2)(1+x-x^2)). a(n)=2*A094966(n) = A000045(n+2)-A039834(n-1). - R. J. Mathar, Oct 30 2008
a(n) = -a(-1-n) for all n in Z. - Michael Somos, Nov 01 2016
a(n) = 2*A000045(ceiling(n/2)*2). - Alois P. Heinz, Nov 02 2016

a(21)-a(22) corrected by Matthew House, Nov 01 2016

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

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A056014 a(n) = (Fibonacci(2n-1) - Fibonacci(n+1))/2.

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A124377 Riordan array (1/(1-x-x^2),x/(1+x)).

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A140833 Sum of Fibonacci numbers between F(-n)....F(n), inclusive.

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