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A138573 a(n) = 2*a(n-1) + 2*a(n-2) + 2*a(n-3) - a(n-4); a(0)=0, a(1)=1, a(2)=2, a(3)=5.

%I A138573 #72 Feb 17 2025 16:05:14
%S A138573 0,1,2,5,16,45,130,377,1088,3145,9090,26269,75920,219413,634114,
%T A138573 1832625,5296384,15306833,44237570,127848949,369490320,1067846845,
%U A138573 3086134658,8919094697,25776662080,74495936025,215297250946,622220603405
%N A138573 a(n) = 2*a(n-1) + 2*a(n-2) + 2*a(n-3) - a(n-4); a(0)=0, a(1)=1, a(2)=2, a(3)=5.
%C A138573 This is a divisibility sequence; that is, if n divides m, then a(n) divides a(m). - _T. D. Noe_, Dec 23 2008
%C A138573 Case P1 = 2, P2 = -4, Q = 1 of the 3-parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - _Peter Bala_, Mar 04 2014
%H A138573 G. C. Greubel, <a href="/A138573/b138573.txt">Table of n, a(n) for n = 0..1000</a>
%H A138573 Kunle Adegoke, Robert Frontczak, and Taras Goy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Goy/goy12.html">Binomial Fibonacci sums from Chebyshev polynomials</a>, J. Integer Seq., Vol. 26 (2023), Art. 23.9.6. See page 4; <a href="https://arxiv.org/abs/2308.04567">Preprint</a>, arXiv:2308.04567 [math.CO], 2023.
%H A138573 Peter Bala, <a href="/A100047/a100047.pdf">Linear divisibility sequences and Chebyshev polynomials</a>
%H A138573 H. C. Williams and R. K. Guy, <a href="http://www.emis.de/journals/INTEGERS/papers/a17self/a17self.Abstract.html">Some Monoapparitic Fourth Order Linear Divisibility Sequences</a>, Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
%H A138573 <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H A138573 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,2,-1).
%F A138573 a(n) = round(w^n/2/sqrt(5)) where w = (1+r)/(1-r) = 2.89005363826396... and r = sqrt(sqrt(5)-2) = 0.485868271756...; for n >= 3, a(n) = A071101(n+3).
%F A138573 G.f.: -x*(x-1)*(1+x)/(1 - 2*x - 2*x^2 - 2*x^3 + x^4). - _R. J. Mathar_, Jun 03 2009
%F A138573 From _Peter Bala_, Mar 04 2014: (Start)
%F A138573 Define a Lucas sequence {U(n)} in the ring of Gaussian integers by the recurrence U(n) = (1 + i)*U(n-1) + U(n-2) with U(0) = 0 and U(1) = 1. Then a(n) = |U(n)|^2.
%F A138573 Let a, b denote the zeros of x^2 - (1 + i)*x - 1 and c, d denote the zeros of x^2 - (1 - i)*x - 1.
%F A138573 Then a(n) = (a^n - b^n)*(c^n - d^n)/((a - b)*(c - d)).
%F A138573 a(n) = (alpha(1)^n + beta(1)^n - alpha(2)^n - beta(2)^n)/(2*sqrt(5)), where alpha(1), beta(1) are the roots of x^2 - ( 1 + sqrt(5))*x + 1 = 0, and alpha(2), beta(2) are the roots of x^2 - (1 - sqrt(5))*x + 1 = 0.
%F A138573 The o.g.f. is the Hadamard product of the rational functions x/(1 - (1 + i)x - x^2) and x/(1 - (1 - i)x - x^2). (End)
%F A138573 From _Peter Bala_, Mar 24 2014: (Start)
%F A138573 a(n) = (1/sqrt(5))*(T(n,phi) - T(n,-1/phi)), where phi = 1/2*(1 + sqrt(5)) is the golden ratio and T(n,x) denotes the Chebyshev polynomial of the first kind. Compare with the Fibonacci numbers, A000045, whose terms are given by the Binet formula 1/sqrt(5)*( phi^n - (-1/phi)^n ).
%F A138573 a(n) = top right (or bottom left) entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, 1; 1, 1]; the off-diagonal elements of M^n give the sequence of Fibonacci numbers. Bottom right entry of the matrix T(n, M) gives A138574. See the remarks in A100047 for the general connection between Chebyshev polynomials and linear divisibility sequences of the fourth order. (End)
%F A138573 a(n) = (((phi + sqrt(phi))^n + (phi - sqrt(phi))^n)/2 - (-1)^n * cos(n*arctan(sqrt(phi))))/sqrt(5), where phi=(1+sqrt(5))/2. - _Vladimir Reshetnikov_, May 11 2016
%F A138573 a(n) = A143056(n+1)^2 + A272665(n+1)^2. - _Vladimir Reshetnikov_, Oct 05 2016
%F A138573 Limit_{n -> oo} a(n)/a(n-1) = A318605. - _A.H.M. Smeets_, Sep 12 2018
%p A138573 seq(coeff(series((x*(1-x)*(x+1))/(1-2*x-2*x^2-2*x^3+x^4),x,n+1), x, n), n = 0 .. 30); # _Muniru A Asiru_, Sep 12 2018
%t A138573 Round@Table[(((GoldenRatio + Sqrt[GoldenRatio])^n + (GoldenRatio - Sqrt[GoldenRatio])^n)/2 - (-1)^n Cos[n ArcTan[Sqrt[GoldenRatio]]])/Sqrt[5], {n, 0, 20}] (* or *) LinearRecurrence[{2, 2, 2, -1}, {0, 1, 2, 5}, 20] (* _Vladimir Reshetnikov_, May 11 2016 *)
%t A138573 Table[Abs[Fibonacci[n, 1 + I]]^2, {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 05 2016 *)
%t A138573 CoefficientList[Series[-x*(x-1)*(1+x)/(1-2*x-2*x^2-2*x^3+x^4), {x, 0, 20}], x] (* _Stefano Spezia_, Sep 12 2018 *)
%o A138573 (PARI) my(x='x+O('x^50)); concat([0], Vec(x*(1-x)*(1+x)/(1 -2*x -2*x^2 -2*x^3 +x^4))) \\ _G. C. Greubel_, Aug 08 2017
%o A138573 (GAP) a:=[0,1,2,5];; for n in [5..30] do a[n]:=2*a[n-1]+2*a[n-2]+2*a[n-3]-a[n-4]; od; a; # _Muniru A Asiru_, Sep 12 2018
%Y A138573 Cf. A071101, A000045, A100047, A138574, A143056, A272665.
%K A138573 nonn
%O A138573 0,3
%A A138573 _Benoit Cloitre_, May 12 2008