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) = 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
KEYWORD
nonn,easy
AUTHOR
Asher Auel, Jun 06 2000
STATUS
approved