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A100329
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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.
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3
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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
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OFFSET
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0,6
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COMMENTS
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Reflected tetranacci numbers (see 1st formula).
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LINKS
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FORMULA
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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).
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
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MAPLE
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a:= n-> (<<1|1|0|0>, <1|0|1|0>, <1|0|0|1>, <1|0|0|0>>^(-n))[1, 4]:
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MATHEMATICA
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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 *)
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PROG
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(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
(SageMath)
@CachedFunction
if (n<4): return (0, 1, -1, 0)[n]
else: return -a(n-1)-a(n-2)-a(n-3)+a(n-4)
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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