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%I #34 Mar 08 2024 12:11:44
%S 1,13,69,409,2377,13861,80781,470833,2744209,15994429,93222357,
%T 543339721,3166815961,18457556053,107578520349,627013566049,
%U 3654502875937,21300003689581,124145519261541,723573111879673
%N Expansion of g.f. (1+8*x-x^2)/((1+x)*(1-6*x+x^2)).
%C A floretion-generated sequence relating the squares of the numerators of continued fraction convergents to sqrt(2) to the squares of the denominators of continued fraction convergents to sqrt(2) (Pell numbers).
%C Floretion Algebra Multiplication Program, FAMP Code:
%C 1dia[J]tesseq[ - .5'j + .5'k - .5j' + .5k' - 2'ii' + 'jj' - 'kk' + .5'ij' + .5'ik' + .5'ji' + 'jk' + .5'ki' + 'kj' + e ]. Identity used: dia[I]tes + dia[J]tes + dia[K]tes = jes + fam + 3tes.
%H G. C. Greubel, <a href="/A105058/b105058.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,5,-1)
%F a(n) = 2 * A001109(n+1) - (-1)^n.
%F G.f.: G(0)/(1-3*x) - 1/(1+x), where G(k) = 1 + 1/(1 - x*(8*k-9)/( x*(8*k-1) - 3/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Aug 12 2013
%F From _G. C. Greubel_, Aug 21 2022: (Start)
%F a(n) = A000129(2*n+2) - (-1)^n.
%F E.g.f.: exp(3*x)*( 2*cosh(2*sqrt(2)*x) + (3/sqrt(2))*sinh(2*sqrt(2)*x)) - exp(-x). (End)
%t CoefficientList[ Series[(1+8x-x^2)/((1+x)(1-6x+x^2)), {x,0,30}], x] (* _Robert G. Wilson v_, Apr 06 2005 *)
%t LinearRecurrence[{5,5,-1}, {1,13,69}, 30] (* _Harvey P. Dale_, Jun 03 2017 *)
%o (Magma) [Evaluate(DicksonSecond(2*n+1, -1), 2) -(-1)^n: n in [0..30]]; // _G. C. Greubel_, Aug 21 2022
%o (SageMath) [lucas_number1(2*n+2,2,-1) -(-1)^n for n in (0..30)] # _G. C. Greubel_, Aug 21 2022
%Y Cf. A000129, A001109, A001333, A046729, A077444.
%Y Cf. A077445, A079291, A082639, A084158, A090390.
%K nonn,easy
%O 0,2
%A _Creighton Dement_, Apr 04 2005
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