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A054492
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a(n) = 3*a(n-1) - a(n-2), a(0)=1, a(1)=6.
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4
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1, 6, 17, 45, 118, 309, 809, 2118, 5545, 14517, 38006, 99501, 260497, 681990, 1785473, 4674429, 12237814, 32039013, 83879225, 219598662, 574916761, 1505151621, 3940538102, 10316462685, 27008849953, 70710087174, 185121411569, 484654147533, 1268841031030
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OFFSET
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0,2
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REFERENCES
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A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196.
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 0..1000
I. Adler, Three Diophantine equations - Part II, Fib. Quart., 7 (1969), pp. 181-193.
E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (3,-1).
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FORMULA
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a(n) = (6*(((3+sqrt(5))/2)^n - ((3-sqrt(5))/2)^n) - (((3+sqrt(5))/2)^(n-1) - ((3-sqrt(5))/2)^(n-1)))/sqrt(5).
a(n) = 2*Lucas(2*n+1) - Fibonacci(2*n+1).
G.f.: (1+3*x)/(1-3*x+x^2). - Philippe Deléham, Nov 03 2008
a(n) = 5*Fibonacci(2*n) + Fibonacci(2*n-1). - Ehren Metcalfe, Mar 26 2016
E.g.f.: (1/10) * exp((3-sqrt(5))*x/2) * ((5-9*sqrt(5)) + (5+9*sqrt(5)) * exp(sqrt(5)*x) ). - G. C. Greubel, Mar 26 2016
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MATHEMATICA
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CoefficientList[Series[(1 + 3 x) / (1 - 3 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 20 2015 *)
LinearRecurrence[{3, -1}, {1, 6}, 100] (* G. C. Greubel, Mar 26 2016 *)
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PROG
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(PARI) Vec((1+3*x)/(1-3*x+x^2) + O(x^30)) \\ Michel Marcus, Mar 20 2015
(MAGMA) I:=[1, 6]; [n le 2 select I[n] else 3*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Mar 20 2015
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CROSSREFS
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Cf. A002878, A054486.
Sequence in context: A262297 A048746 A026382 * A128525 A083334 A199113
Adjacent sequences: A054489 A054490 A054491 * A054493 A054494 A054495
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KEYWORD
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easy,nonn
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AUTHOR
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Barry E. Williams, May 06 2000
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EXTENSIONS
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More terms from Vincenzo Librandi, Mar 20 2015
Typo in name fixed by Karl V. Keller, Jr., Jun 23 2015
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STATUS
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approved
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