|
|
A052993
|
|
a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3), with a(0)=a(1)=1, a(2)=4.
|
|
4
|
|
|
1, 1, 4, 4, 13, 13, 40, 40, 121, 121, 364, 364, 1093, 1093, 3280, 3280, 9841, 9841, 29524, 29524, 88573, 88573, 265720, 265720, 797161, 797161, 2391484, 2391484, 7174453, 7174453, 21523360, 21523360, 64570081, 64570081, 193710244
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
LINKS
|
|
|
FORMULA
|
G.f.: 1/((1-3*x^2)*(1-x)).
a(n+2) = 3*a(n) + 1, where a(0) = a(1) = 1.
a(n) = -1/2 + Sum((1/4)*(1+3*_alpha)*_alpha^(-1-n), _alpha = RootOf(-1 + 3*_Z^2)).
a(n) = Sum{k=0..n} 3^(k/2)*(1-(-1)^k)/(2*sqrt(3)). - Paul Barry, Jul 28 2004
|
|
MAPLE
|
spec := [S, {S=Prod(Sequence(Prod(Union(Z, Z, Z), Z)), Sequence(Z))}, unlabeled ]: seq(combstruct[count ](spec, size=n), n=0..20);
|
|
MATHEMATICA
|
LinearRecurrence[{1, 3, -3}, {1, 1, 4}, 30] (* or *) RecurrenceTable[{a[n + 2] == 3*a[n] + 1, a[0] == 1, a[1] == 1}, a, {n, 0, 30}] (* G. C. Greubel, Nov 21 2018 *)
|
|
PROG
|
(PARI) x='x+O('x^30); Vec(1/((1-3*x^2)*(1-x))) \\ G. C. Greubel, Nov 21 2018
(Magma) I:=[1, 1, 4]; [n le 3 select I[n] else Self(n-1) +3*Self(n-2) -3*Self(n-3): n in [1..30]]; // G. C. Greubel, Nov 21 2018
(Sage) s=(1/((1-3*x^2)*(1-x))).series(x, 30); s.coefficients(x, sparse=False) # G. C. Greubel, Nov 21 2018
|
|
CROSSREFS
|
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|