cp's OEIS Frontend

A085504 Horadam sequence (0,1,9,3).

%I A085504 #29 Feb 16 2025 08:32:50
%S A085504 0,1,18,81,405,1944,9477,45927,223074,1082565,5255361,25509168,
%T A085504 123825753,601059771,2917611090,14162371209,68745613437,333698181192,
%U A085504 1619805064509,7862698824255,38166342053346,185263315578333,899287025215113,4365230915850336
%N A085504 Horadam sequence (0,1,9,3).
%C A085504 Lim_{n->infinity} a(n)/a(n-1) = (3/2)*(1 + sqrt(5)), which can also be written as phi^2 + 2*phi - 1, phi^3 + phi - 1, phi + sqrt(5) + 1, 3*phi, 3*phi^2 - 3, phi^4 - 2 and lim_{n->infinity} (3/2)*(1 + Lucas(n)/Fibonacci(n)).
%H A085504 Eric Weisstein, <a href="https://mathworld.wolfram.com/HoradamSequence.html">Horadam Sequence</a>
%H A085504 Eric Weisstein, <a href="https://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>
%H A085504 Eric Weisstein, <a href="https://mathworld.wolfram.com/PellNumber.html">Pell Number</a>
%H A085504 Eric Weisstein, <a href="https://mathworld.wolfram.com/LucasNumber.html">Lucas Number</a>
%H A085504 Eric Weisstein, <a href="https://mathworld.wolfram.com/LucasSequence.html">Lucas Sequence</a>
%H A085504 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,9).
%F A085504 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.
%F A085504 G.f.: x*(1+15*x+18*x^2)/(1-3*x-9*x^2). [_Colin Barker_, Jun 20 2012]
%e A085504 a(4) = 405 because a(3) = 81, a(2) = 18, s = 3, r = 9 and (3 * 81) + (9 * 18) = 405.
%t A085504 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 *)
%Y A085504 Cf. A024318, A000032, A000129, A001076, A085939.
%Y A085504 Essentially the same as A122069 and A099012.
%K A085504 nonn,easy
%O A085504 0,3
%A A085504 _Ross La Haye_, Aug 18 2003
%E A085504 First formula corrected and more terms from _Harvey P. Dale_, Nov 24 2012