OFFSET
0,9
COMMENTS
a(n+3) is the number of tilings of an n-board (a board with dimensions n X 1) with (1/2,1/2;2)-combs and (1/2,1/2;3)-combs. A (w,g;m)-comb is a tile composed of m pieces of dimensions w X 1 separated horizontally by gaps of width g. - Michael A. Allen, Sep 25 2024
LINKS
Michael A. Allen and Kenneth Edwards, Connections between two classes of generalized Fibonacci numbers squared and permanents of (0,1) Toeplitz matrices, Lin. Multilin. Alg. 72:13 (2024) 2091-2103.
Index entries for linear recurrences with constant coefficients, signature (1,1,1,-1,1,-1).
FORMULA
a(n) = A000931(n)^2.
a(n) = a(n-1) + a(n-2) + a(n-3) - a(n-4) + a(n-5) - a(n-6) for n >= 6.
G.f.: (x^5+x^2+x-1)/(-x^6+x^5-x^4+x^3+x^2+x-1).
a(n) = a(n-2) + a(n-3) + 2*Sum_{r=8..n} ( A000930(r-8)*a(n+3-r) ). - Michael A. Allen, Sep 25 2024
EXAMPLE
a(10)=9 because Padovan(10)=3 and 3^2=9.
MATHEMATICA
a[0] = a[3] = a[5] = a[6] = 1; a[1] = a[2] = a[4] = 0; a[n_Integer] := a[n] = 2*a[n - 2] + 2*a[n - 3] - a[n - 7]; Table[a[i], {i, 0, 40}] (* Olivier Gérard, Jul 05 2011 *)
Table[RootSum[-1 - # + #^3 &, #^n (5 - 6 # + 4 #^2) &]^2/529, {n, 0,
40}] (* Eric W. Weisstein, Apr 16 2018 *)
LinearRecurrence[{1, 1, 1, -1, 1, -1}, {1, 0, 0, 1, 0, 1}, 40] (* Eric W. Weisstein, Apr 16 2018 *)
PROG
(PARI) Vec(O(x^20)+(1-x-x^2-x^5)/(1-x-x^2-x^3+x^4-x^5+x^6)) \\ Charles R Greathouse IV, Jul 05 2011
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Omar E. Pol, Nov 02 2007
STATUS
editing