OFFSET
1,2
LINKS
Eric Babson and Einar Steingrimsson, Generalized Permutation Patterns and Classification of the Mahonian Statistics, Séminaire Lotharingien de Combinatoire, B44b (2000), 18 pp.
Index entries for linear recurrences with constant coefficients, signature (15, -28, 15, -1).
FORMULA
a(1) = 1; a(2) = 4; a(3) = 35; a(4) = 428; a(n) = 15a(n-1)-28a(n-2)+15a(n-3)-a(n-4) for n>= 5. The minimal polynomial of M is x^4-15x^3+28x^2-15x+1, the coefficients of which yield the coefficients of the recurrence relation.
O.g.f.: x*(1-11*x+3*x^2)/((x^2-13*x+1)*(x-1)^2). - R. J. Mathar, Dec 05 2007
MAPLE
a[1]:=1:a[2]:=4:a[3]:=35:a[4]:=428: for n from 5 to 18 do a[n]:=15*a[n-1]-28*a[n-2]+15*a[n-3]-a[n-4] od: seq(a[n], n=1..18);
with(linalg): M[1]:=matrix(4, 4, [1, 1, 1, 1, 0, 1, 2, 3, 0, 1, 3, 6, 0, 1, 4, 10]): for n from 2 to 17 do M[n]:=multiply(M[1], M[n-1]) od: 1, seq(sum(M[n][1, i], i=1..4), n=1..17);
MATHEMATICA
M = {{ 1, 1, 1, 1 }, { 0, 1, 2, 3 }, {0, 1, 3, 6 }, { 0, 1, 4, 10 }}; v[1] = {1, 1, 1, 1}; v[n_] := v[n] = M.v[n - 1]; a1 = Table[v[n][[1]], {n, 1, 25}]
LinearRecurrence[{15, -28, 15, -1}, {1, 4, 35, 428}, 50] (* Vincenzo Librandi, Jun 15 2015 *)
PROG
(Magma) I:=[1, 4, 35, 428]; [n le 4 select I[n] else 15*Self(n-1)-28*Self(n-2)+15*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jun 15 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Roger L. Bagula, Oct 05 2006
EXTENSIONS
Edited by N. J. A. Sloane, Oct 15 2006
STATUS
approved