OFFSET
0,4
COMMENTS
The first three of the sequence of polynomials: x^n-(x+1)^n are Pisots, suggesting the name Pascal-Pisots as the (x+1)^n polynomials give the Pascal triangular sequence. r = Abs[Table[x /. NSolve[Det[M - IdentityMatrix[4]*x] == 0, x][[n]], {n, 1, 4}]] gives:{0.5497, 0.831739, 0.831739, 2.62966}
LINKS
Index entries for linear recurrences with constant coefficients, signature (1,3,3,1).
FORMULA
M = {{0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {1, 3, 3, 1}}; w[0] = {0, 1, 1, 2}; w[n_] := w[n] = M.w[n - 1] a(n) = w[n][[1]].
a(n) = a(n-1)+3*a(n-2)+3*a(n-3)+a(n-4). G.f.: x*(2*x^2 -1)/(x^4 +3*x^3 +3*x^2 +x -1). [Colin Barker, Oct 11 2012]
MATHEMATICA
M = {{0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {1, 3, 3, 1}}; w[0] = {0, 1, 1, 2}; w[n_] := w[n] = M.w[n - 1] a = Flatten[Table[w[n][[1]], {n, 0, 25}]]
LinearRecurrence[{1, 3, 3, 1}, {0, 1, 1, 2}, 30] (* Harvey P. Dale, Jan 13 2015 *)
PROG
(PARI) concat(0, Vec((1 - 2*x^2)/(1 - x - 3*x^2 - 3*x^3 - x^4)+O(x^99))) \\ Charles R Greathouse IV, Oct 11 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Roger L. Bagula, Feb 18 2006
STATUS
approved