A193885 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) - a(n-4), n>=4; a(0) = 1, a(1) = 1, a(2) = 2, a(3) = 3.

1, 1, 2, 3, 3, 1, -5, -18, -41, -75, -115, -143, -118, 35, 431, 1213, 2499, 4254, 6047, 6665, 3609, -7375, -32334, -77933, -147781, -234503, -305765, -283634, -20329, 718653, 2239077, 4824577, 8495482, 12533139, 14698471, 10166901, -9557053, -57006530

Offset: 0

Author

Johannes W. Meijer, Aug 11 2011

Keywords

• sign
• easy

Comments

The Ze1 sums, see A180662, of triangle A108299 equal the terms of this sequence.

Links

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

Crossrefs

Cf. A099531, A108299, A180662.

Programs

Magma

I:=[1, 1, 2, 3 ]; [n le 4 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 10 2012

Maple

A193885 := proc(n) option remember: if n=0 then 1 elif n=1 then 1 elif n=2 then 2 elif n=3 then 3 elif n>=4 then 3*procname(n-1)-3*procname(n-2)+procname(n-3)-procname(n-4) fi: end: seq(A193885(n),n=0..37);

Mathematica

CoefficientList[Series[(1-x)*(1-x+x^2)/(1-3*x+3*x^2-x^3+x^4),{x,0,50}],x] (* Vincenzo Librandi, Jul 10 2012 *)

Formula

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) - a(n-4), n>=4; a(0) = 1, a(1) = 1, a(2) = 2, a(3) = 3.

G.f.: (1-x)*(1-x+x^2)/(1-3*x+3*x^2-x^3+x^4).

a(n) = (-1)^(n+1)*(A099531(n+4) + 2*A099531(n+3) + 2*A099531(n+2) + A099531(n+1)).