|
| |
|
|
A052899
|
|
G.f.: ( 1-2*x ) / ((x-1)*(4*x^2+2*x-1)).
|
|
3
| |
|
|
1, 1, 5, 13, 45, 141, 461, 1485, 4813, 15565, 50381, 163021, 527565, 1707213, 5524685, 17878221, 57855181, 187223245, 605867213, 1960627405, 6344723661, 20531956941, 66442808525, 215013444813, 695798123725, 2251650026701
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,3
|
|
|
COMMENTS
| Contribution by L. Edson Jeffery, Apr 19 2011. (Start):
Let A be the unit-primitive matrix (see [Jeffery])
A=A_(10,4)=
(0 0 0 0 1)
(0 0 0 2 0)
(0 0 2 0 1)
(0 2 0 2 0)
(2 0 2 0 1).
Then a(n)=(1/5)*Trace(A^n). (End)
|
|
|
LINKS
| INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 875
L. E. Jeffery, Unit-primitive matrices
Index to sequences with linear recurrences with constant coefficients, signature (3,2,-4).
|
|
|
FORMULA
| Recurrence: {a(1)=1, a(0)=1, -4*a(n)-2*a(n+1)+a(n+2)+1 =0}
Sum(-1/25*(-1-8*_alpha+4*_alpha^2)*_alpha^(-1-n), _alpha=RootOf(1-3*_Z-2*_Z^2+4*_Z^3))
a(n)/a(n-1) tends to (1 + sqrt(5)) = 3.236067... - Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 01 2008
a(n)=(1/5)*Sum_{k=1..5} ((x_k)^4-3*(x_k)^2+1), x_k=2*cos((2*k-1)*Pi/10). Also, a(n)/a(n-1) -> spectral radius of matrix A_(10,4) above. - L. Edson Jeffery, Apr 19 2011
a(n) = (2*A087131(n)+1)/5. - Bruno Berselli, Apr 20 2011
|
|
|
MAPLE
| spec := [S, {S=Sequence(Prod(Union(Sequence(Union(Z, Z)), Z, Z), Z))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
|
|
|
PROG
| (Sage) from sage.combinat.sloane_functions import recur_gen2b
it = recur_gen2b(1, 1, 2, 4, lambda n:-1)
[it.next() for i in xrange(1, 28)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 09 2008
(MAGMA) [(1/5)*(2^(n+1)*Lucas(n)+1): n in [0..50]]; // Vincenzo Librandi, Apr 20 2011
(Maxima) makelist(coeff(taylor((1-2*x)/(1-3*x-2*x^2+4*x^3), x, 0, n), x, n), n, 0, 25); [Bruno Berselli, May 30 2011]
|
|
|
CROSSREFS
| Cf. A084057.
Sequence in context: A115785 A113835 A006349 * A147200 A147396 A099972
Adjacent sequences: A052896 A052897 A052898 * A052900 A052901 A052902
|
|
|
KEYWORD
| easy,nonn,changed
|
|
|
AUTHOR
| encyclopedia(AT)pommard.inria.fr, Jan 25 2000
|
|
|
EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jun 08 2000
|
| |
|
|
