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A180148
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a(n) = 3*a(n-1) + a(n-2) with a(0)=2 and a(1)=5.
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5
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2, 5, 17, 56, 185, 611, 2018, 6665, 22013, 72704, 240125, 793079, 2619362, 8651165, 28572857, 94369736, 311682065, 1029415931, 3399929858, 11229205505, 37087546373, 122491844624, 404563080245, 1336181085359, 4413106336322, 14575500094325, 48139606619297
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OFFSET
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0,1
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COMMENTS
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Inverse binomial transform of A052961 (without the leading 1).
For n >= 1, also the number of matchings in the n-alkane graph. - Eric W. Weisstein, Jul 14 2021
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LINKS
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Eric Weisstein's World of Mathematics, Matching
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FORMULA
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G.f.: (2-x)/(1-3*x-x^2).
a(n) = 3*a(n-1) + a(n-2) with a(0)=2 and a(1)=5.
a(n) = ((4+7*A)*A^(-n-1) + (4+7*B)*B^(-n-1))/13 with A = (-3+sqrt(13))/2 and B = (-3-sqrt(13))/2.
Lim_{k->infinity} a(n+k)/a(k) = (-1)^n*2/(A006497(n) - A006190(n)*sqrt(13)).
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MAPLE
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a:= n-> (<<0|1>, <1|3>>^n. <<2, 5>>)[1, 1]:
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MATHEMATICA
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CoefficientList[Series[(2 - x)/(1 - 3 x - x^2), {x, 0, 20}], x] (* Eric W. Weisstein, Jul 14 2021 *)
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PROG
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CROSSREFS
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Cf. A000602 (more information on n-alkanes).
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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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