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 A054492 a(n) = 3*a(n-1) - a(n-2), a(0)=1, a(1)=6. 4
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 REFERENCES A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196. LINKS 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). FORMULA 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 MATHEMATICA 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 *) PROG (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 CROSSREFS Cf. A002878, A054486. Sequence in context: A262297 A048746 A026382 * A128525 A083334 A199113 Adjacent sequences:  A054489 A054490 A054491 * A054493 A054494 A054495 KEYWORD easy,nonn AUTHOR Barry E. Williams, May 06 2000 EXTENSIONS More terms from Vincenzo Librandi, Mar 20 2015 Typo in name fixed by Karl V. Keller, Jr., Jun 23 2015 STATUS approved

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