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Expansion of (1+x)*(2*x-1)/((1-x)*(x^2+2*x-1)).
1

%I #24 Dec 18 2023 12:20:52

%S 1,2,3,6,13,30,71,170,409,986,2379,5742,13861,33462,80783,195026,

%T 470833,1136690,2744211,6625110,15994429,38613966,93222359,225058682,

%U 543339721,1311738122,3166815963,7645370046,18457556053,44560482150,107578520351

%N Expansion of (1+x)*(2*x-1)/((1-x)*(x^2+2*x-1)).

%C Pisano period lengths: 1, 2, 8, 4, 12, 8, 6, 8, 24, 12, 24, 8, 28, 6, 24, 16, 16, 24, 40, 12,.. (is this A175181?) - _R. J. Mathar_, Aug 10 2012

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-1).

%F a(n) = 2*a(n-1) + a(n-2) - 2, with a(0)=1, a(1)=2.

%F From _R. J. Mathar_, Mar 17 2010: (Start)

%F a(n) = A052937(n-1), n > 0.

%F a(n) = 3*a(n-1) - a(n-2) - a(n-3). (End)

%e a(2) = 2*a(1) + a(0) - 2 = 2*2 + 1 - 2 = 3

%e a(3) = 2*a(2) + a(1) - 2 = 2*3 + 2 - 2 = 6.

%t LinearRecurrence[{3, -1, -1}, {1, 2, 3}, 31] (* _Robert P. P. McKone_, Apr 03 2022 *)

%Y Cf. A174192, A001333 (first differences).

%K nonn

%O 0,2

%A _Clark Kimberling_, Mar 11 2010