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

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

0,5

Comments

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

References

H. Eriksson and M. Jonsson, Level sizes of the Bulgarian solitaire game tree, Fib. Q., 35:3 (2017), 243-251.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

É. Czabarka, R. Flórez, L. Junes, A Discrete Convolution on the Generalized Hosoya Triangle, Journal of Integer Sequences, 18 (2015), #15.1.6.

Index entries for linear recurrences with constant coefficients, signature (4,-3,-2,1)

Formula

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

Mathematica

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 *)

Prog

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

Crossrefs

Cf. A000045, A056015.

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

Sequence in context: A084851 A094706 A325927 * A247287 A159036 A058693

Adjacent sequences: A056011 A056012 A056013 * A056015 A056016 A056017

Keyword

nonn,easy

Author

Asher Auel (asher.auel(AT)reed.edu), Jun 06 2000

Status

approved