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

Links

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).

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

a(n) = Sum_{k=1..n-1} binomial(n-1, k) * A094966(k-1) (Othsuka, 2024). - Amiram Eldar, Feb 29 2024

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, A000079, A051450, A056015, A094966.

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, Jun 06 2000

Status

approved