OFFSET
0,3
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..300
R. Bacher and D. Garber, Spindle-configurations of skew lines, Geom. Topol 11 (2007) 1049.
FORMULA
G.f.: ( 3*(1-z)-sqrt((1-z)*(1-5*z-4*z^2)) ) / (2*(1-z)).
a(n) = 2 + Sum_{k=1..n-1} a(n-k)*a(k). - Benoit Cloitre, Jul 27 2008
Recurrence: n*a(n) = 2*(3*n-4)*a(n-1) - (n+2)*a(n-2) - 2*(2*n-7)*a(n-3). - Vaclav Kotesovec, Oct 18 2012
a(n) ~ sqrt(41-3*sqrt(41))*((5+sqrt(41))/2)^n/(16*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 18 2012
a(n) = Sum_{k=1..n} (binomial(2*k-2, k-1)*Sum_{i=0..n-k} binomial(k, n-k-i)*binomial(k+i-1, k-1)/k), n > 0, a(0)=1. - Vladimir Kruchinin, Jan 24 2013
a(n+1) starting (1, 3, ...) = (first n terms) dot product (first n difference terms), added to a(n). - Gary W. Adamson, May 20 2013
EXAMPLE
a(5) = 104. (1, 3, 8, 27) dot (1, 2, 5, 19) = 77; then 104 = a(4) + 77 = 27 + 77.
MAPLE
G:=(3*(1-z)-sqrt((1-z)*(1-5*z-4*z^2)))/2/(1-z): Gser:=series(G, z=0, 32): seq(coeff(Gser, z, n), n=0..27); # Emeric Deutsch, Dec 31 2006
MATHEMATICA
CoefficientList[Series[(3*(1-x)-Sqrt[(1-x)*(1-5*x-4*x^2)])/2/(1-x), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 18 2012 *)
a[n_] := Sum[(Binomial[2*k-2, k-1]*Sum[Binomial[k, n-k-i]*Binomial[k+i-1, k-1], {i, 0, n-k}])/k, {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Jan 24 2013, after Vladimir Kruchinin *)
PROG
(Maxima) a(n):=sum((binomial(2*k-2, k-1)*sum(binomial(k, n-k-i)*binomial(k+i-1, k-1), i, 0, n-k))/k, k, 1, n); /* Vladimir Kruchinin, Jan 24 2013 */
(PARI)
N = 66; x = 'x + O('x^N);
gf = ( 3*(1-x)-sqrt((1-x)*(1-5*x-4*x^2)) ) / (2*(1-x));
v = Vec(gf)
/* Joerg Arndt, Jan 24 2013 */
CROSSREFS
KEYWORD
nonn
AUTHOR
David Garber, Sep 19 2005
EXTENSIONS
More terms from Emeric Deutsch, Dec 31 2006
STATUS
approved