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%I #25 Nov 02 2019 08:46:27
%S 1,4,17,80,369,1700,7841,36160,166753,768996,3546289,16354000,
%T 75417809,347795396,1603886913,7396455680,34109360321,157298104900,
%U 725393076049,3345209499600,15426707209777,71141522037604,328074947492321,1512944453384000,6977067089461281
%N Norm of coefficients in the expansion of 1/(1 - 2*x - i*x^2), where i is the imaginary unit.
%C The radius of convergence of g.f. equals 1 + sqrt(2) - sqrt(2)*sqrt(1 + sqrt(2)) = 0.216845335...
%C The following remarks assume an offset of 1. This sequence is a divisibility sequence, i.e., a(n) divides a(m) whenever n divides m. The sequence satisfies a linear recurrence of order 4. It is the case P1 = 4, P2 = -4, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - _Peter Bala_, Mar 25 2014
%H Peter Bala, <a href="/A100047/a100047.pdf">Linear divisibility sequences and Chebyshev polynomials</a>
%H H. C. Williams and R. K. Guy, <a href="http://dx.doi.org/10.1142/S1793042111004587">Some fourth-order linear divisibility sequences</a>, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
%H H. C. Williams and R. K. Guy, <a href="https://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 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,2,4,-1).
%F G.f.: (1 - x^2)/(1 - 4*x - 2*x^2 - 4*x^3 + x^4).
%F From _Peter Bala_, Mar 25 2014: (Start)
%F The following formulas assume an offset of 1.
%F a(n) = 1/(2*sqrt(2))*(T(n,1 + sqrt(2)) - T(n,1 - sqrt(2))), where T(n,x) denotes the Chebyshev polynomial of the first kind.
%F a(n) = the bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, 1; 1, 2]. Note, the bottom left element of the matrix M^n gives the Lucas sequence A000129.
%F a(n) = U(n-1,exp(2*i*Pi/8))*U(n-1,exp(-2*i*Pi/8)) = U(n-1,(1 + i)/sqrt(2))*U(n-1,(1 - i)/sqrt(2)), where U(n,x) denotes the Chebyshev polynomial of the second kind.
%F The o.g.f. is the Chebyshev transform of the rational function x/(1 - 4*x - 4*x^2), where the Chebyshev transform takes the function A(x) to the function (1 - x^2)/(1 + x^2)*A(x/(1 + x^2)). See the remarks in A100047 for the general connection between Chebyshev polynomials and 4th-order linear divisibility sequences. (End)
%e G.f.: A(x) = 1 + 4*x + 17*x^2 + 80*x^3 + 369*x^4 + 1700*x^5 + 7841*x^6 +...
%e The terms equal the norm of the complex coefficients in the expansion:
%e 1/(1 - 2*x - i*x^2) = 1 + 2*x + (4 + i)*x^2 + (8 + 4*i)*x^3 + (15 + 12*i)*x^4 + (26 + 32*i)*x^5 + (40 + 79*i)*x^6 + (48 + 184*i)*x^7 +...
%e so that
%e a(1) = 2^2, a(2) = 4^2 + 1, a(3) = 8^2 + 4^2, a(4) = 15^2 + 12^2, a(5) = 26^2 + 32^2, ...
%t LinearRecurrence[{4, 2, 4, -1}, {1, 4, 17, 80}, 25] (* _Jean-François Alcover_, Nov 02 2019 *)
%o (PARI) {a(n)=norm(polcoeff(1/(1-2*x-I*x^2+x*O(x^n)), n))}
%o for(n=0,31,print1(a(n),", "))
%Y Cf. A105309, A218135.
%Y Cf. also A000129, A057087, A100047.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Oct 21 2012
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