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A062092
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a(n) = 2*a(n-1) - (-1)^n for n > 0, a(0)=2.
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13
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2, 5, 9, 19, 37, 75, 149, 299, 597, 1195, 2389, 4779, 9557, 19115, 38229, 76459, 152917, 305835, 611669, 1223339, 2446677, 4893355, 9786709, 19573419, 39146837, 78293675, 156587349, 313174699, 626349397, 1252698795, 2505397589
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OFFSET
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0,1
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COMMENTS
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Let A be the Hessenberg matrix of order n, defined by: A[1,j] = A[i,i] = 1, A[i,i-1] = -1, and A[i,j] = 0 otherwise. Then, for n>=1, a(n-1) = charpoly(A,3). - Milan Janjic, Jan 24 2010
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REFERENCES
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T. Koshy, Fibonacci and Lucas numbers with applications, Wiley, 2001, p. 98.
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LINKS
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FORMULA
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a(n) = a(n-1) + 2*a(n-2).
a(n) = (7*2^n - (-1)^n)/3.
G.f.: (2+3*x)/(1-x-2*x^2).
E.g.f.: (7*exp(2*x) - exp(-x))/3.
a(n) = Sum_{j=0..2} A001045(n-j) (sum of 3 consecutive elements of the Jacobsthal sequence). - Alexander Adamchuk, May 16 2006
a(n) = A005009(n-1) - a(n-1) for n >= 1.
a(n) = a(n-2) + A005009(n-2) for n >= 2.
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MATHEMATICA
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PROG
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(Magma) [(7*2^n-(-1)^n)/3: n in [0..40]]; // G. C. Greubel, Apr 04 2023
(SageMath) [(7*2^n-(-1)^n)/3 for n in range(41)] # G. C. Greubel, Apr 04 2023
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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