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%I #73 Mar 16 2024 02:37:56
%S 5,81,1445,25921,465125,8346321,149768645,2687489281,48225038405,
%T 865363202001,15528312597605,278644263554881,5000068431390245,
%U 89722587501469521,1610006506595061125,28890394531209630721,518417095055178291845,9302617316461999622481
%N Expansion of (5-14*x+x^2)/((1-x)*(x^2-18*x+1)).
%C Relates squares of numerators and denominators of continued fraction convergents to sqrt(5).
%C 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)
%C Floretion Algebra Multiplication Program, FAMP Code: tesseq[A*B*C].
%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
%H G. C. Greubel, <a href="/A115032/b115032.txt">Table of n, a(n) for n = 0..795</a>
%H Creighton Dement, <a href="https://github.com/Floretion-Inquisitor/floretions/blob/main/examples/A115032/A115032.floretions.pdf">Floretions associated with A115032</a>.
%H Wolfdieter Lang, <a href="/A115032/a115032_5.pdf">A proof for the touching circle problem (part I)</a>.
%H Giovanni Lucca, <a href="https://ijgeometry.com/product/giovanni-lucca-circle-chains-inside-the-arbelos-and-integer-sequences/">Circle chains inside the arbelos and integer sequences</a>, Int'l J. Geom. (2023) Vol. 12, No. 1, 71-82.
%H Kival Ngaokrajang, <a href="/A115032/a115032_2.pdf">Illustration of initial terms</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (19,-19,1).
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%F sqrt(a(2*n)) = sqrt(5)*A007805(n) = sqrt(5)*Fibonacci(6*n+3)/2 = sqrt(5)*A001076(2*n+1); sqrt(a(2*n+1)) = A023039(2*n+1) = A001077(2*n).
%F From _Wolfdieter Lang_, Aug 22 2014: (Start)
%F O.g.f.: (5-14*x+x^2)/((1-x)*(x^2-18*x+1)) (see the name).
%F 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.
%F a(n) = (9*S(n, 18) - S(n-1, 18) + 1)/2, with the Chebyshev S-polynomials (see A049310). From A049660.
%F a(n) = (A023039(n+1) + 1)/2.
%F (End)
%F a(n) = 19*a(n-1) - 19*a(n-2) + a(n-3). - _Colin Barker_, Aug 23 2014
%F From _Wolfdieter Lang_, Aug 24 2014: (Start)
%F 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.
%F 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)).
%F (See the _Colin Barker_ formula from Aug 04 2014 in the history of A240926.) (End)
%e G.f. = 5 + 81*x + 1445*x^2 + 25921*x^3 + 465125*x^4 + 8346321*x^5 + ...
%p seq((9*combinat:-fibonacci(6*(n+1)) - combinat:-fibonacci(6*n) + 8)/16, n = 0 .. 20); # _Robert Israel_, Aug 25 2014
%t LinearRecurrence[{19,-19,1},{5,81,1445},30] (* _Harvey P. Dale_, Nov 14 2014 *)
%t CoefficientList[Series[(5 - 14*x + x^2)/((1 - x)*(x^2 - 18*x + 1)), {x, 0, 50}], x] (* _G. C. Greubel_, Dec 19 2017 *)
%o (PARI) Vec((5-14*x+x^2)/((1-x)*(x^2-18*x+1)) + O(x^20)) \\ _Michel Marcus_, Aug 23 2014
%Y Cf. A001076, A001077, A007805, A023039, A097924.
%Y Cf. also A000045, A049660, A049310, A023039. - _Wolfdieter Lang_, Aug 22 2014
%K easy,nonn
%O 0,1
%A _Creighton Dement_, Feb 26 2006
%E More terms from _Michel Marcus_, Aug 23 2014
%E Edited (comment by _Kival Ngaokrajang_ rewritten, Chebyshev index link added) by _Wolfdieter Lang_, Aug 26 2014
%E Partially edited by _Jon E. Schoenfield_ and _N. J. A. Sloane_, Mar 15 2024
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