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 A105058 Expansion of g.f. (1+8*x-x^2)/((1+x)*(1-6*x+x^2)). 1
 1, 13, 69, 409, 2377, 13861, 80781, 470833, 2744209, 15994429, 93222357, 543339721, 3166815961, 18457556053, 107578520349, 627013566049, 3654502875937, 21300003689581, 124145519261541, 723573111879673 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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). LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (5,5,-1) FORMULA a(n) = 2 * A001109(n+1) - (-1)^n. 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 From G. C. Greubel, Aug 21 2022: (Start) a(n) = A000129(2*n+2) - (-1)^n. E.g.f.: exp(3*x)*( 2*cosh(2*sqrt(2)*x) + (3/sqrt(2))*sinh(2*sqrt(2)*x)) - exp(-x). (End) MATHEMATICA CoefficientList[ Series[(1+8x-x^2)/((1+x)(1-6x+x^2)), {x, 0, 30}], x] (* Robert G. Wilson v, Apr 06 2005 *) LinearRecurrence[{5, 5, -1}, {1, 13, 69}, 30] (* Harvey P. Dale, Jun 03 2017 *) PROG Floretion Algebra Multiplication Program, FAMP Code: 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. (Magma) [Evaluate(DicksonSecond(2*n+1, -1), 2) -(-1)^n: n in [0..30]]; // G. C. Greubel, Aug 21 2022 (SageMath) [lucas_number1(2*n+2, 2, -1) -(-1)^n for n in (0..30)] # G. C. Greubel, Aug 21 2022 CROSSREFS Cf. A000129, A001109, A001333, A046729, A077444. Cf. A077445, A079291, A082639, A084158, A090390. Sequence in context: A137188 A055338 A055880 * A146469 A146381 A085461 Adjacent sequences: A105055 A105056 A105057 * A105059 A105060 A105061 KEYWORD nonn,easy AUTHOR Creighton Dement, Apr 04 2005 STATUS approved

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Last modified December 5 10:23 EST 2022. Contains 358586 sequences. (Running on oeis4.)