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%I #41 Sep 25 2023 07:24:52
%S -2,1,-6,16,-62,225,-842,3136,-11706,43681,-163022,608400,-2270582,
%T 8473921,-31625106,118026496,-440480882,1643897025,-6135107222,
%U 22896531856,-85451020206,318907548961,-1190179175642,4441809153600,-16577057438762,61866420601441,-230888624967006
%N Expansion of g.f. (2+7*x+2*x^2)/((x^2-1)*(1+4*x+x^2)).
%C One of 4 related sequences. This is the sequence "les(n)". "jes(n)" = [1, -4, 15, -56, ...] is (-1)^(n+1)*A001353(n+1), "tes(n)" is A097948 and "ves(n)" is A099949.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-4,0,4,1).
%F Properties (from _Creighton Dement_, Sep 06 2004):
%F I: jes(n) + les(n) + tes(n) = ves(n)
%F II: All of the following are perfect squares: {les(2n+1); tes(2n+1); ves(2n+1); ves(2n+1) - jes(2n+1) - 1 = les(2n+1) + tes(2n+1) - 1; 3*les(2n+1) + 1 = 3*jes(n)^2 + 1}.
%F III: les(2n+1) divides ves(2n+1) - jes(2n+1) - 1 = les(2n+1) + tes(2n+1) - 1
%F IV: (jes(n))^2 = les(2n+1)
%F V: tes(2n) = A001570(n), sqrt( tes(2n+1) ) = A001075(n)
%F VI: sqrt( ves(2n+1) ) = A001835(n)
%F VII: sqrt( les(2n+1) ) = A001353(n)
%F VIII: les(n) + tes(n) = ves(2+n) + jes(n)
%F IX: lim n |jes(n+1)/jes(n)| = lim n |les(n+1)/les(n)| = lim n |tes(n+1)/tes(n)| = lim n |ves(n+1)/ves(n)| = 2 + sqrt(3)
%F Comment from Roland Bacher, Sep 07 2004: These 4 sequences satisfy jes(n+1)=-4*jes(n)-jes(n-1), les(n+1)=les(n-1)+jes(n), ves(n+1)=les(n-1)-jes(n-1)+tes(n-1), tes(n+1)=les(n-1)+3*jes(n), plus initial conditions for n=0, 1.
%F 12*a(n) = -11 -9*(-1)^n -2*(-1)^n*A001075(n+1). - _R. J. Mathar_, May 21 2019
%F From _Eric Simon Jacob_, Aug 26 2023: (Start)
%F a(n) = ( ( sqrt(3) - 2 )^(n+1) + ( -sqrt(3) - 2 )^(n+1) + 9*(-1)^(n+1) - 11 )/12.
%F a(n) = ( 2*cosh( (n+1)*log(sqrt(3) - 2) ) + 9*(-1)^(n+1) - 11 )/12. (End)
%t LinearRecurrence[{-4, 0, 4, 1}, {-2, 1, -6, 16}, 27] (* _Robert P. P. McKone_, Aug 25 2023 *)
%Y Cf. A001353, A097948, A097949.
%K sign,easy
%O 0,1
%A _N. J. A. Sloane_, following a suggestion of _Creighton Dement_, Sep 06 2004
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