A115032 Expansion of (5-14*x+x^2)/((1-x)*(x^2-18*x+1)).

5, 81, 1445, 25921, 465125, 8346321, 149768645, 2687489281, 48225038405, 865363202001, 15528312597605, 278644263554881, 5000068431390245, 89722587501469521, 1610006506595061125, 28890394531209630721, 518417095055178291845, 9302617316461999622481

Offset: 0

Author

Creighton Dement, Feb 26 2006

Keywords

• easy
• nonn

Comments

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

Examples

			G.f. = 5 + 81*x + 1445*x^2 + 25921*x^3 + 465125*x^4 + 8346321*x^5 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..795
• Creighton Dement, Floretions associated with A115032.
• 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.
• Kival Ngaokrajang, Illustration of initial terms.
• Index entries for linear recurrences with constant coefficients, signature (19,-19,1).
• Index entries for sequences related to Chebyshev polynomials.

Crossrefs

Cf. A001076, A001077, A007805, A023039, A097924.

Cf. also A000045, A049660, A049310, A023039. - Wolfdieter Lang, Aug 22 2014

Programs

Maple

seq((9*combinat:-fibonacci(6*(n+1)) - combinat:-fibonacci(6*n) + 8)/16, n = 0 .. 20); # Robert Israel, Aug 25 2014

Mathematica

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 *)

PARI

Vec((5-14*x+x^2)/((1-x)*(x^2-18*x+1)) + O(x^20)) \\ Michel Marcus, Aug 23 2014

Formula

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).

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)

Extensions

More terms from Michel Marcus, Aug 23 2014

Edited (comment by Kival Ngaokrajang rewritten, Chebyshev index link added) by Wolfdieter Lang, Aug 26 2014

Partially edited by Jon E. Schoenfield and N. J. A. Sloane, Mar 15 2024