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A163606
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a(n) = ((3 + 2*sqrt(2))*(3 + sqrt(2))^n + (3 - 2*sqrt(2))*(3 - sqrt(2))^n)/2.
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3
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3, 13, 57, 251, 1107, 4885, 21561, 95171, 420099, 1854397, 8185689, 36133355, 159500307, 704068357, 3107907993, 13718969459, 60558460803, 267317978605, 1179998646009, 5208766025819, 22992605632851, 101494271616373
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OFFSET
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0,1
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COMMENTS
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For n >= 1, a(n-1) is the number of generalized compositions of n when there are 2^(i-1)+1 different types of i, (i=1,2,...). - Milan Janjic, Sep 24 2010
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LINKS
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FORMULA
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a(n) = 6*a(n-1) - 7*a(n-2) for n > 1; a(0) = 3, a(1) = 13.
G.f.: (3-5*x)/(1-6*x+7*x^2).
E.g.f.: exp(3*x)*( 3*cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Jul 29 2017
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MATHEMATICA
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LinearRecurrence[{6, -7}, {3, 13}, 40] (* Harvey P. Dale, Dec 24 2011 *)
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PROG
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(Magma) Z<x>:= PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((3+2*r)*(3+r)^n+(3-2*r)*(3-r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 07 2009
(PARI) x='x+O('x^50); Vec((3-5*x)/(1-6*x+7*x^2)) \\ G. C. Greubel, Jul 29 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Al Hakanson (hawkuu(AT)gmail.com), Aug 01 2009
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EXTENSIONS
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STATUS
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approved
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