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A131300
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a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4) with n>3, a(0)=1, a(1)=2, a(2)=3, a(3)=7.
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4
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1, 2, 3, 7, 14, 27, 49, 86, 147, 247, 410, 675, 1105, 1802, 2931, 4759, 7718, 12507, 20257, 32798, 53091, 85927, 139058, 225027, 364129, 589202, 953379, 1542631, 2496062, 4038747, 6534865, 10573670, 17108595, 27682327, 44790986, 72473379, 117264433, 189737882
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OFFSET
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0,2
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COMMENTS
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a(n)/a(n-1) tends to phi.
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LINKS
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FORMULA
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G.f.: (1-x-x^2+3*x^3)/((1-x-x^2)*(1-x)^2). [Bruno Berselli, May 03 2012]
a(n) = (3*A131269(n)-n-1))/2 = 3*((1+sqrt(5))^(n+2)-(1-sqrt(5))^(n+2))/(2^(n+2)*sqrt(5))-2*(n+1). [Bruno Berselli, May 03 2012]
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EXAMPLE
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a(4) = 14 = (1 + 1 + 7 + 4 + 1).
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MAPLE
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seq(add(3*binomial(floor((n+k)/2), k)-2, k=0..n), n=0..50); # Nathaniel Johnston, Jun 29 2011
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MATHEMATICA
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LinearRecurrence[{3, -2, -1, 1}, {1, 2, 3, 7}, 38] (* Bruno Berselli, May 03 2012 *)
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PROG
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(PARI) Vec((1-x-x^2+3*x^3)/((1-x-x^2)*(1-x)^2)+O(x^38))
(Magma) /* By first comment: */ [&+[3*Binomial(n-Floor((k+1)/2), Floor(k/2))-2: k in [0..n]]: n in [0..37]];
(Maxima) makelist(expand(3*((1+sqrt(5))^(n+2)-(1-sqrt(5))^(n+2))/(2^(n+2)*sqrt(5))-2*(n+1)), n, 0, 37); (End)
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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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