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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)
a(n-1), n >=0, with a(-1) = 1, is also the circle curvature of circles inscribed in a special way in the larger segment of a circle of radius 5/4 (in some length units) divided by a chord of length 2. When considering the smaller segment, a similar circle curvature sequence will be given in A240926. For more details see comments on A240926. See the illustration in the link, 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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G. C. Greubel, Table of n, a(n) for n = 0..795
Kival Ngaokrajang, Illustration of initial terms
Wolfdieter Lang, A proof for the touching circle problem (part I).
Giovanni Lucca, Circle chains inside the arbelos and integer sequences, Int'l J. Geom. (2023) Vol. 12, No. 1, 71-82.
Index entries for linear recurrences with constant coefficients, signature (19,-19,1).
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
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sqrt(a(2*n)) = sqrt(5)*A007805(n) = sqrt(5)*Fib(6*n+3)/2 = sqrt(5)*A001076(2*n+1). sqrt(a(2*n+1)) = A023039(2*n+1) = A001077(2*n).
From Wolfdieter Lang, Aug 22 2014: (Start)
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.
a(n) = (A023039(n+1) + 1)/2.
(End)
a(n) = 19*a(n-1)-19*a(n-2)+a(n-3). - Colin Barker, Aug 23 2014
From Wolfdieter Lang, Aug 24 2014: (Start)
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)).
(See the Colin Barker formula from Aug 04 2014 in the history of A240926.) (End)
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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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Floretion Algebra Multiplication Program, FAMP Code: tesseq[A*B*C] (see comment).
(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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Cf. A001076, A001077, A007805, A023039, A097924.
Cf. A000045, A049660, A049310, A023039. - Wolfdieter Lang, Aug 22 2014
Sequence in context: A110257 A335177 A135918 * A278883 A307376 A009733
Adjacent sequences: A115029 A115030 A115031 * A115033 A115034 A115035
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KEYWORD
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easy,nonn
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AUTHOR
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Creighton Dement, Feb 26 2006
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EXTENSIONS
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More terms from Michel Marcus, Aug 23 2014
Edited: comment by Kival Ngaokrajang rewritten. Chebyshev index link added. - Wolfdieter Lang, Aug 26 2014
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STATUS
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approved
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