A012781 Take every 5th term of Padovan sequence A000931, beginning with the second term.

0, 1, 4, 16, 65, 265, 1081, 4410, 17991, 73396, 299426, 1221537, 4983377, 20330163, 82938844, 338356945, 1380359512, 5631308624, 22973462017, 93722435101, 382349636061, 1559831901918, 6363483400447, 25960439030624

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Number of nonisomorphic graded posets with 0 and uniform hasse graph of rank n, with exactly 2 elements of each rank level above 0, for n > 0. (Uniform used in the sense of Retakh, Serconek and Wilson.) Here, we do not assume all maximal elements have maximal rank and thus use graded poset to mean: For every element x, all maximal chains among those with x as greatest element have the same finite length. - David Nacin, Feb 13 2012

References

• R. Stanley, Enumerative combinatorics, Vol. 1, Cambridge University Press, Cambridge, 1997, pp. 96-100.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• V. Retakh, S. Serconek, and R. Wilson, Hilbert Series of Algebras Associated to Directed Graphs and Order Homology, arXiv:1010.6295 [math.RA], 2010-2011.
• Index entries for linear recurrences with constant coefficients, signature (5,-4,1).

Programs

Magma

I:=[0, 1, 4 ]; [n le 3 select I[n] else 5*Self(n-1)-4*Self(n-2)+Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 03 2012

Mathematica

LinearRecurrence[{5, -4, 1}, {0, 1, 4}, 25] (* Harvey P. Dale, Jan 10 2012 *)

Python

def a(n, adict={0:0, 1:1, 2:4}):
    if n in adict:
        return adict[n]
    adict[n]=5*a(n-1) - 4*a(n-2) + a(n-3)
    return adict[n] # David Nacin, Feb 27 2012

Formula

a(n+3) = 5*a(n+2) - 4*a(n+1) + a(n).

G.f.: x*(1-x)/(1-5*x+4*x^2-x^3). - Colin Barker, Feb 03 2012

Extensions

Initial term 0 added by Colin Barker, Feb 03 2012