OFFSET
0,3
COMMENTS
a(n+1) is also the number of sequences of length 2n obeying the regular expression "0^* (1 or 2)^* 3^*" and having sum 3n. For example, a(3)=11 because of the sequences 0033, 0123, 0213, 0222, 1113, 1122, 1212, 1221, 2112, 2121, 2211. - Don Knuth, May 11 2016
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..300
FORMULA
a(n+1) - 4*a(n) = -A081696(n-1).
From Vaclav Kotesovec, Oct 21 2012: (Start)
G.f.: x*(x-sqrt(1-4*x))/(sqrt(1-4*x)*(x^2+4*x-1)).
Recurrence: (n-2)*a(n) = 2*(4*n-9)*a(n-1) - (15*n-38)*a(n-2) - 2*(2*n-5)* a(n-3).
a(n) ~ 4^n/sqrt(Pi*n). (End)
a(n) = Sum_{k=1..n} (F(k)*binomial(2*n-k-1,n-k)), where F(k) = A000045(k). - Vladimir Kruchinin, Mar 17 2016
Simpler g.f.: x/sqrt(1-4*x)/(x+sqrt(1-4*x)). - Don Knuth, May 11 2016
a(n) = 4*a(n-1) + a(n-2) - binomial(2*n-4,n-2) for n>=2. - Hrishikesh Venkataraman, Jul 02 2021
MAPLE
with(combinat); seq( add(binomial(2*n-k-1, n-k)*fibonacci(k), k=0..n), n=0..30); # G. C. Greubel, Nov 28 2019
1/(sqrt(1 - 4*x) + 1/x - 4): series(%, x, 27):
seq(coeff(%, x, k), k=0..26); # Peter Luschny, May 29 2021
MATHEMATICA
t[n_, k_]:= CoefficientList[ Series[x/(1-x-x^2)/(1-x)^k, {x, 0, k}], x][[k+1]]; Array[ t[#, #] &, 20]
Table[Sum[Binomial[2*n-k-1, n-k]*Fibonacci[k], {k, 0, n}], {n, 0, 30}] (* G. C. Greubel, Nov 28 2019 *)
PROG
(Maxima)
a(n):=sum(fib(k)*binomial(2*n-k-1, n-k), k, 1, n); /* Vladimir Kruchinin, Mar 17 2016 */
(PARI) a(n) = sum(k=1, n, fibonacci(k)*binomial(2*n-k-1, n-k)) \\ Michel Marcus, Mar 17 2016
(Magma) [(&+[Binomial(2*n-k-1, n-k)*Fibonacci(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Nov 28 2019
(Sage) [sum(binomial(2*n-k-1, n-k)*fibonacci(k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Nov 28 2019
(GAP) List([0..30], n-> Sum([0..n], k-> Binomial(2*n-k-1, n-k)*Fibonacci(k) )); # G. C. Greubel, Nov 28 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Curtz, Apr 08 2010
STATUS
approved