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First differences of A052911.
2

%I #14 Feb 13 2022 04:36:07

%S 1,5,14,45,139,434,1351,4209,13110,40837,127203,396226,1234207,

%T 3844441,11975078,37301261,116189979,361921042,1127350583,3511592833,

%U 10938286998,34071752661,106130359315,330586256610

%N First differences of A052911.

%C a(n)/a(n-1) tends to 3.11490754148...an eigenvalue of M and a root of the characteristic polynomial x^3 - 3x^2 - x + 2.

%D Boris A. Bondarenko, "Generalized Pascal Triangles and Pyramids, Their Fractals, Graphs and Applications"; Fibonacci Association, 1993, p. 27.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,-2).

%F G.f.: (2*x^2-2*x-1)*x / (-2*x^3+x^2+3*x-1).

%F Recurrence: a(n) = 3*a(n-1) + a(n-2) - 2*a(n-3).

%F a(n) = rightmost term in M^5 * [1 1 1], where M = the 3 X 3 upper triangular matrix [2 1 2 / 1 1 0 / 1 0 0].

%F INVERT transform of (1, 4, 5, 6, 7, 8, 9, ...) with offset 0.

%e a(5) = 139 = rightmost term in M^5 * [1 1 1] which is [434 205 139]. 434 = a(6), while 205 = A052911(5).

%e a(6) = 434 = 3*a(5) + a(4) - 2*a(3) = 3*139 + 45 - 2*14.

%t LinearRecurrence[{3,1,-2},{1,5,14},30] (* _Harvey P. Dale_, Apr 21 2016 *)

%Y Cf. A019481, A052550, A052939, A100058, A058071.

%K nonn

%O 1,2

%A _Gary W. Adamson_, Oct 31 2004

%E Edited by _Ralf Stephan_, Nov 02 2004