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A115032
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Expansion of (5-14*x+x^2)/((1-x)*(x^2-18*x+1)).
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8
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5, 81, 1445, 25921, 465125, 8346321, 149768645, 2687489281, 48225038405, 865363202001, 15528312597605, 278644263554881, 5000068431390245, 89722587501469521, 1610006506595061125, 28890394531209630721, 518417095055178291845, 9302617316461999622481
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OFFSET
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0,1
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COMMENTS
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Relates squares of numerators and denominators of continued fraction convergents to sqrt(5).
Sequence is generated by the floretion A*B*C with A = + 'i - 'k + i' - k' - 'jj' - 'ij' - 'ji' - 'jk' - 'kj' ; B = - 'i + 'j - i' + j' - 'kk' - 'ik' - 'jk' - 'ki' - 'kj' ; C = - 'j + 'k - j' + k' - 'ii' - 'ij' - 'ik' - 'ji' - 'ki' (apart from a factor (-1)^n)
Floretion Algebra Multiplication Program, FAMP Code: tesseq[A*B*C].
The sequence a(n-1), n >= 0, with a(-1) = 1, is also the curvature of circles inscribed in a special way in the larger segment of a circle of radius 5/4 (in some units) cut by a chord of length 2. For the smaller segment, the analogous curvature sequence is given in A240926. For more details see comments on A240926. See also the illustration in the present sequence, and the proof of the coincidence of the curvatures with a(n-1) in part I of the W. Lang link. - Kival Ngaokrajang, Aug 23 2014
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LINKS
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FORMULA
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O.g.f.: (5-14*x+x^2)/((1-x)*(x^2-18*x+1)) (see the name).
a(n) = (9*F(6*(n+1)) - F(6*n) + 8)/16, n >= 0 with F(n) = A000045(n) (Fibonacci). From the partial fraction decomposition of the o.g.f.: (1/2)*((9 - x)/(1 - 18*x + x^2) + 1/(1 - x)). For F(6*n)/8 see A049660(n). a(-1) = 1 with F(-6) = -F(6) = -8.
a(n) = (9*S(n, 18) - S(n-1, 18) + 1)/2, with the Chebyshev S-polynomials (see A049310). From A049660.
(End)
a(n) = 19*a(n-1) - 19*a(n-2) + a(n-3). - Colin Barker, Aug 23 2014
a(n) = 18*a(n-1) - a(n-2) - 8, n >= 1, a(-1) = 1, a(0) = 5. See the Chebyshev S-polynomial formula above.
The o.g.f. for the sequence a(n-1) with a(-1) = 1, n >= 0, [1, 5, 81, 1445, ..] is (1-14*x+5*x^2)/((1-x)*(1-18*x+x^2)).
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EXAMPLE
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G.f. = 5 + 81*x + 1445*x^2 + 25921*x^3 + 465125*x^4 + 8346321*x^5 + ...
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MAPLE
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seq((9*combinat:-fibonacci(6*(n+1)) - combinat:-fibonacci(6*n) + 8)/16, n = 0 .. 20); # Robert Israel, Aug 25 2014
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MATHEMATICA
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LinearRecurrence[{19, -19, 1}, {5, 81, 1445}, 30] (* Harvey P. Dale, Nov 14 2014 *)
CoefficientList[Series[(5 - 14*x + x^2)/((1 - x)*(x^2 - 18*x + 1)), {x, 0, 50}], x] (* G. C. Greubel, Dec 19 2017 *)
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PROG
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(PARI) Vec((5-14*x+x^2)/((1-x)*(x^2-18*x+1)) + O(x^20)) \\ Michel Marcus, Aug 23 2014
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CROSSREFS
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KEYWORD
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easy,nonn,changed
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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