A070778 Let M denote the 6 X 6 matrix = row by row /1,1,1,1,1,1/1,1,1,1,1,0/1,1,1,1,0,0/1,1,1,0,0,0/1,1,0,0,0,0/1,0,0,0,0,0/ and A(n) the vector (x(n),y(n),z(n),t(n),u(n),v(n)) = M^n*A where A is the vector (1,1,1,1,1,1); then a(n) = u(n).

1, 2, 11, 41, 176, 721, 3003, 12439, 51623, 214103, 888173, 3684174, 15282475, 63393324, 262962987, 1090800411, 4524765831, 18769248040, 77856998326, 322959774150, 1339674254489, 5557122741105, 23051583675890, 95620617831960, 396645310086831, 1645330322871807

Offset: 0

Author

Henry Bottomley, May 06 2002

Keywords

• easy
• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,6,-4,-5,1,1).

Crossrefs

Cf. A006359, A069007, A069008, A069009, A070778, A006359 (offset), for x(n), y(n), z(n), t(n), u(n), v(n).

Programs

Magma

I:=[1,2,11,41,176,721]; [n le 6 select I[n] else 3*Self(n-1)+6*Self(n-2)-4*Self(n-3)-5*Self(n-4)+Self(n-5)+Self(n-6): n in [1..30]]; // Vincenzo Librandi, Oct 10 2017

Maple

a:= n-> (Matrix(6, (i, j)->`if`(i+j>7, 0, 1))^n.<<[1$6][]>>)[5, 1]:
seq(a(n), n=0..30);  # Alois P. Heinz, Jun 14 2013

Mathematica

CoefficientList[Series[(x^2 + x - 1)/(x^6 + x^5 - 5*x^4 - 4*x^3 + 6*x^2 + 3*x - 1), {x, 0, 30}], x] (* Wesley Ivan Hurt, Oct 09 2017 *)
LinearRecurrence[{3, 6, -4, -5, 1, 1}, {1, 2, 11, 41, 176, 721}, 30] (* Vincenzo Librandi, Oct 10 2017 *)

Formula

a(n) = 2*A006359(n-1) - A006359(n-3) for n > 2.

G.f.: (x^2 + x - 1) / (x^6 + x^5 - 5*x^4 - 4*x^3 + 6*x^2 + 3*x - 1). - Colin Barker, Jun 14 2013

a(n) = 3*a(n-1) + 6*a(n-2) - 4*a(n-3) - 5*a(n-4) + a(n-5) + a(n-6). - Wesley Ivan Hurt, Oct 09 2017