A100329 a(n) = -a(n-1) -a(n-2) -a(n-3) +a(n-4), a(0)=0, a(1)=1, a(2)=-1, a(3)=0.

0, 1, -1, 0, 0, 2, -3, 1, 0, 4, -8, 5, -1, 8, -20, 18, -7, 17, -48, 56, -32, 41, -113, 160, -120, 114, -267, 433, -400, 348, -648, 1133, -1233, 1096, -1644, 2914, -3599, 3425, -4384, 7472, -10112, 10449, -12193, 19328, -27696, 31010, -34835, 50849, -74720, 89716, -100680

Offset: 0

Author

Mitch Harris, Nov 16 2004

Keywords

• sign
• easy

Comments

Reflected tetranacci numbers (see 1st formula).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Kai Wang, Identities, generating functions and Binet formula for generalized k-nacci sequences, 2020.
• Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1,1).

Crossrefs

Cf. A000078.

Cf. A000073 (tribonacci), A057597 (reflected tribonacci).

Programs

Magma

I:=[0,1,-1,0]; [n le 4 select I[n] else -Self(n-1) -Self(n-2) -Self(n-3) +Self(n-4): n in [1..61]]; // G. C. Greubel, Jan 30 2023

Maple

a:= n-> (<<1|1|0|0>, <1|0|1|0>, <1|0|0|1>, <1|0|0|0>>^(-n))[1, 4]:
seq(a(n), n=0..50);  # Alois P. Heinz, Jun 12 2008

Mathematica

CoefficientList[Series[x/(1+x+x^2+x^3-x^4), {x, 0, 50}], x]
LinearRecurrence[{-1,-1,-1,1},{0,1,-1,0},60] (* Harvey P. Dale, May 20 2018 *)

SageMath

@CachedFunction
def a(n): # a=A100329
    if (n<4): return (0,1,-1,0)[n]
    else: return -a(n-1)-a(n-2)-a(n-3)+a(n-4)
[a(n) for n in range(61)] # G. C. Greubel, Jan 30 2023

Formula

a(n) = A000078(-n).

Let Q(n) = A000078, then a(n) = (-1)^(n+1)*(Q(n)^3 - 2*Q(n-1)*Q(n) *Q(n+1) + Q(n-2)*Q(n+1)^2 + Q(n-1)^2*Q(n+2) - Qn(-2)*Q(n)*Q(n+2)) derived from powers of the inverse of a generalized Fibonacci matrix.

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

G.f. of absolute values: x/(1-x+x^2-x^3-x^4). - Vaclav Kotesovec, Oct 18 2013

a(n) = term (1,4) in the 4 X 4 matrix [1,1,0,0; 1,0,1,0; 1,0,0,1; 1,0,0,0]^(-n). - Alois P. Heinz, Jun 12 2008