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A122558
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a(0)=1, a(1)=3, a(n) = 4*a(n-1) + 3*a(n-2) for n > 1.
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2
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1, 3, 15, 69, 321, 1491, 6927, 32181, 149505, 694563, 3226767, 14990757, 69643329, 323545587, 1503112335, 6983086101, 32441681409, 150715983939, 700188979983, 3252903871749, 15112182426945, 70207441323027, 326166312572943
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OFFSET
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0,2
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COMMENTS
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a(n) is the number of compositions of n when there are 3 types of 1 and 6 types of other natural numbers. - Milan Janjic, Aug 13 2010
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LINKS
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Table of n, a(n) for n=0..22.
Index entries for linear recurrences with constant coefficients, signature (4,3).
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FORMULA
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G.f.: (1-x)/(1-4*x-3*x^2).
a(n) = Sum_{k=0..n} 3^k*A122542(n,k).
Limit_{n->infinity} a(n+1)/a(n) = 2 + sqrt(7) = 4.645751311064....
a(n) = -(1/14)*(2-sqrt(7))^n*sqrt(7) + (1/14)*sqrt(7)*(2+sqrt(7))^n + (1/2)*(2-sqrt(7))^n + (1/2)*(2+sqrt(7))^n. - Paolo P. Lava, Nov 19 2008
a(n) = ((7+sqrt(7))/14)*(2+sqrt(7))^n + ((7-sqrt(7))/14)*(2-sqrt(7))^n. - Richard Choulet, Nov 20 2008
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PROG
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(PARI) Vec((1-x)/(1-4*x-3*x^2) + O(x^30)) \\ Michel Marcus, Feb 04 2022
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CROSSREFS
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Cf. A122542.
Sequence in context: A217451 A213451 A224749 * A110211 A167874 A318967
Adjacent sequences: A122555 A122556 A122557 * A122559 A122560 A122561
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KEYWORD
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nonn
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AUTHOR
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Philippe Deléham, Sep 20 2006, Sep 22 2006
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EXTENSIONS
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Corrected by T. D. Noe, Nov 07 2006
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STATUS
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approved
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