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A097076
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Expansion of g.f. x/(1 - x - 3*x^2 - x^3).
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13
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0, 1, 1, 4, 8, 21, 49, 120, 288, 697, 1681, 4060, 9800, 23661, 57121, 137904, 332928, 803761, 1940449, 4684660, 11309768, 27304197, 65918161, 159140520, 384199200, 927538921, 2239277041, 5406093004, 13051463048, 31509019101, 76069501249, 183648021600
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OFFSET
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0,4
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COMMENTS
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Counts walks of length n between two vertices of a triangle, when a loop has been added at the third vertex.
a(n) is the center term of the 3 X 3 matrix [0,1,0; 0,0,1; 1,3,1]^n. - Gary W. Adamson, May 30 2008
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LINKS
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FORMULA
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a(n) = ( (1+sqrt(2))^n + (1-sqrt(2))^n - 2*(-1)^n )/4.
a(n) = a(n-1) + 3*a(n-2) + a(n-3). [corrected by Paul Curtz, Mar 04 2008]
a(n) = (Sum_{k=0..floor(n/2)} binomial(n, 2*k)*2^k)/2 - (-1)^n/2.
a(n) = Sum_{k=0..n} (-1)^k*Pell(n-k). - Paul Barry, Oct 22 2009
G.f.: x / ( (1+x)*(1-2*x-x^2) ).
a(n) + a(n+1) = A000129(n+1). (End)
E.g.f.: (exp(x)*cosh(sqrt(2)*x) - cosh(x) + sinh(x))/2. - Stefano Spezia, Mar 31 2024
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MATHEMATICA
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CoefficientList[Series[x/(1-x-3x^2-x^3), {x, 0, 40}], x] (* or *) LinearRecurrence[{1, 3, 1}, {0, 1, 1}, 40] (* Vladimir Joseph Stephan Orlovsky, Jan 30 2012 *)
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PROG
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(Magma) [(Evaluate(DicksonFirst(n, -1), 2) -2*(-1)^n)/4: n in [0..40]]; // G. C. Greubel, Aug 18 2022
(SageMath) [(lucas_number2(n, 2, -1) -2*(-1)^n)/4 for n in (0..40)] # G. C. Greubel, Aug 18 2022
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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