|
|
A099781
|
|
a(n) = Sum_{k=0..floor(n/3)} C(n-k,2*k) * 4^(n-3*k).
|
|
7
|
|
|
1, 4, 16, 65, 268, 1120, 4737, 20244, 87280, 379073, 1656348, 7272896, 32060673, 141775396, 628505296, 2791696705, 12419264300, 55315472416, 246607247233, 1100229683508, 4911436984752, 21934428189121, 97992663440444
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
In general a(n) = Sum_{k=0..floor(n/3)} C(n-k,2*k) * u^k * v^(n-3*k) has g.f. (1-v*x)/((1-v*x)^2 - u*x^2) and satisfies the recurrence a(n) = 2*u*v*a(n-1) - v^2*a(n-2) + u*a(n-3).
|
|
LINKS
|
|
|
FORMULA
|
G.f.: (1-4*x)/((1-4*x)^2 - x^3).
a(n) = 8*a(n-1) - 16*a(n-2) + a(n-3).
|
|
MAPLE
|
seq(coeff(series((1-4*x)/((1-4*x)^2 -x^3), x, n+1), x, n), n = 0..30); # G. C. Greubel, Sep 04 2019
|
|
MATHEMATICA
|
LinearRecurrence[{8, -16, 1}, {1, 4, 16}, 30] (* Harvey P. Dale, Jul 07 2013 *)
|
|
PROG
|
(PARI) my(x='x+O('x^30)); Vec((1-4*x)/((1-4*x)^2 -x^3)) \\ G. C. Greubel, Sep 04 2019
(Magma) I:=[1, 4, 16]; [n le 3 select I[n] else 8*Self(n-1) - 16*Self(n-2) + Self(n-3): n in [1..30]];
(Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P((1-4*x)/((1-4*x)^2 -x^3)).list()
(GAP) a:=[1, 4, 16];; for n in [4..30] do a[n]:=8*a[n-1]-16*a[n-2]+a[n-3]; od; a; # G. C. Greubel, Sep 04 2019
|
|
CROSSREFS
|
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|