OFFSET
0,6
COMMENTS
P(0)=P(1)=P(2)=P(3)=1, for m > 3: P(m) = P(m-3) + P(m-4) is the 3rd sequence in the series: Fibonacci sequence, Padovan sequence, ... The Padovan sequence (whose ratio of successive terms approaches the plastic constant) is similar to the Perrin sequence. - Jonathan Vos Post, Jan 23 2005
Binomial transform yields A079398 without the initial (0,1,1,1). - R. J. Mathar, Apr 09 2008
a(n+1) corresponds to the diagonal sums of "triangle": 1; 1; 1; 1,1; 1,1; 1,1; 1,2,1; 1,2,1; 1,2,1; 1,3,3,1; 1,3,3,1; 1,3,3,1; 1,4,6,4,1; ..., rows of Pascal's triangle (A007318) repeated three times. - Philippe Deléham, Dec 13 2008
a(n) is the number of pairs of rabbits living at month n with the following rules: a pair of rabbits born in month n begins to procreate in month n + 3, procreates again in month n + 4, and dies at the end of this month (each pair therefore gives birth to 2 pairs); warning! The first pair is born in month 2. - Robert FERREOL, Oct 24 2017
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Vedran Krcadinac, A new generalization of the golden ratio, Fibonacci Quart. 44 (2006), no. 4, 335-340.
Eric Weisstein's World of Mathematics, Padovan Sequence.
Index entries for linear recurrences with constant coefficients, signature (0,0,1,1).
FORMULA
a(0)=0, a(1)=1, a(2)=1, a(3)=1, a(n) = a(n-3) + a(n-4) for n > 3. - Colin Barker, Sep 18 2013
From Paul Barry, Jul 06 2004: (Start)
a(n) = Sum_{k=0..floor((n-1)/2)} binomial(floor((n-k-1)/3), k) (offset 0).
a(n) = Sum_{k=0..floor(n/2)} binomial(floor((n-k-1)/3), k)}-0^n (offset 0). (End)
For n > 1, a(n) = P(n-2) where P(n) is defined by: P(0)=P(1)=P(2)=P(3)=1, for m > 3: P(m) = P(m-3) + P(m-4). - Jonathan Vos Post, Jan 23 2005
The same sequence may be constructed as follows: Let M = {{0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {1, 1, 0, 0}}; v[1] = {1, 1, 1, 1}; v[n] = M.v[n - 1]. Then a(n) = v[n][[1]]. - Roger L. Bagula, Sep 16 2006
O.g.f.: -x^2*(1+x+x^2)/(-1+x^3+x^4). a(n) = A017817(n-1) + A017817(n-2) + A017817(n-3). - R. J. Mathar, Apr 09 2008
MATHEMATICA
CoefficientList[Series[x (1 + x + x^2)/(1 - x^3 - x^4), {x, 0, 60}], x] (* Vincenzo Librandi, Mar 16 2014 *)
LinearRecurrence[{0, 0, 1, 1}, {0, 1, 1, 1}, 60] (* Jean-François Alcover, Dec 05 2017 *)
nxt[{a_, b_, c_, d_}]:={b, c, d, a+b}; NestList[nxt, {0, 1, 1, 1}, 60][[;; , 1]] (* Harvey P. Dale, Apr 27 2023 *)
PROG
(PARI) a(n)=([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; 1, 1, 0, 0]^n*[0; 1; 1; 1])[1, 1] \\ Charles R Greathouse IV, Oct 03 2016
(PARI) x='x+O('x^50); concat([0], Vec(x*(1+x+x^2)/(1-x^3-x^4))) \\ G. C. Greubel, Apr 30 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Benoit Cloitre, Feb 16 2003
EXTENSIONS
Recurrence corrected by Colin Barker, Sep 18 2013
STATUS
approved