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A100295
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Simple recursive sequence generated from a symmetric matrix.
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2
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1, 3, 14, 61, 269, 1184, 5213, 22951, 101046, 444873, 1958633, 8623232, 37965321, 167149115, 735903870, 3239948389, 14264452181, 62801801632, 276496162501, 1217323801087, 5359485727718, 23596094350545, 103886025056529, 457376803199488, 2013683168560465
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OFFSET
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1,2
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COMMENTS
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The characteristic polynomial of M = x^3 - 4*x^2 - 2*x + 1.
Limit_{n -> oo} a(n)/a(n-1) tends to 4.4026788295..., a root of the polynomial and an eigenvalue of M.
A100296 uses the alternative operation M^n * [1, 1, 1].
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LINKS
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FORMULA
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a(n) = rightmost term in M^n * [1, 0, 0], where M = [{3, 2, 1}, {2, 1, 0}, {1, 0, 0}].
a(n) = 4*a(n-1) + 2*a(n-2) - a(n-3).
G.f.: x*(1-x)/(1-4*x-2*x^2+x^3). - Colin Barker, May 25 2013
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EXAMPLE
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a(4) = 61 since M^4 * [1, 0, 0] = [269, 158, 61]. (leftmost term = a(5). M
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MAPLE
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a:= n-> (<<3|2|1>, <2|1|0>, <1|0|0>>^n)[1, 3]:
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MATHEMATICA
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LinearRecurrence[{4, 2, -1}, {1, 3, 14}, 40] (* G. C. Greubel, Feb 05 2023 *)
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PROG
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(Magma) I:=[1, 3, 14]; [n le 3 select I[n] else 4*Self(n-1) +2*Self(n-2) -Self(n-3): n in [1..40]]; // G. C. Greubel, Feb 05 2023
(SageMath)
@CachedFunction
if (n<3): return (0, 1, 3)[n]
else: return 4*a(n-1) + 2*a(n-2) - a(n-3)
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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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