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 A085504 Horadam sequence (0,1,9,3). 1
 0, 1, 18, 81, 405, 1944, 9477, 45927, 223074, 1082565, 5255361, 25509168, 123825753, 601059771, 2917611090, 14162371209, 68745613437, 333698181192, 1619805064509, 7862698824255, 38166342053346, 185263315578333, 899287025215113, 4365230915850336 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS a(n) / a(n-1) converges to (3 + (3 * 5^1/2)) / 2 as n approaches infinity. (3 + (3 * 5^1/2)) / 2 can also be written as Phi^2 + (2 * Phi) - 1, Phi^3 + Phi - 1, Phi + 5^1/2 + 1, 3 * Phi, (3 * Phi^2) - 3, Phi^4 - 2 and (3 + (3 * (L(n) / F(n)))) / 2, where L(n) is the n-th Lucas number and F(n) is the n-th Fibonacci number as n approaches infinity. LINKS Eric Weisstein, Horadam Sequence Eric Weisstein, Fibonacci Number Eric Weisstein, Pell Number Eric Weisstein, Lucas Number Eric Weisstein, Lucas Sequence Index entries for linear recurrences with constant coefficients, signature (3,9). FORMULA a(n) = s*a(n-1) + r*a(n-2); for n > 3, where a(0) = 0, a(1) = 1, a(2) = 18, a(4) = 81, s = 3, r = 9. G.f.: x*(1+15*x+18*x^2)/(1-3*x-9*x^2). [Colin Barker, Jun 20 2012] EXAMPLE a(4) = 405 because a(3) = 81, a(2) = 18, s = 3, r = 9 and (3 * 81) + (9 * 18) = 405. MATHEMATICA Join[{0, 1}, LinearRecurrence[{3, 9}, {18, 81}, 30]] (* or *) CoefficientList[ Series[x (1+15x+18x^2)/(1-3x-9x^2), {x, 0, 30}], x] (* Harvey P. Dale, Nov 24 2012 *) CROSSREFS Cf. A024318, A000032, A000129, A001076, A085939. Sequence in context: A039408 A043231 A044011 * A214531 A271502 A235641 Adjacent sequences:  A085501 A085502 A085503 * A085505 A085506 A085507 KEYWORD nonn,easy AUTHOR Ross La Haye, Aug 18 2003 EXTENSIONS First formula corrected and more terms from_Harvey P. Dale_, Nov 24 2012 STATUS approved

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Last modified August 24 03:20 EDT 2019. Contains 326260 sequences. (Running on oeis4.)