login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A115032 Expansion of (5-14*x+x^2)/((1-x)*(x^2-18*x+1)). 8
5, 81, 1445, 25921, 465125, 8346321, 149768645, 2687489281, 48225038405, 865363202001, 15528312597605, 278644263554881, 5000068431390245, 89722587501469521, 1610006506595061125, 28890394531209630721, 518417095055178291845, 9302617316461999622481 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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)

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

LINKS

Table of n, a(n) for n=0..17.

Kival Ngaokrajang, Illustration of initial terms

Wolfdieter Lang, A proof for the touching circle problem (part I).

Index entries for linear recurrences with constant coefficients, signature (19,-19,1).

Index entries for sequences related to Chebyshev polynomials.

FORMULA

sqrt(a(2n)) = sqrt(5)*A007805(n) = sqrt(5)*Fib(6n+3)/2 = sqrt(5)*A001076(2n+1). sqrt(a(2n+1)) = A023039(2n+1) = A001077(2n).

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)

EXAMPLE

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

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 *)

PROG

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

CROSSREFS

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

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

Sequence in context: A275347 A110257 A135918 * A278883 A009733 A009756

Adjacent sequences:  A115029 A115030 A115031 * A115033 A115034 A115035

KEYWORD

easy,nonn

AUTHOR

Creighton Dement, Feb 26 2006

EXTENSIONS

More terms from Michel Marcus, Aug 23 2014

Edited: comment by Kival Ngaokrajang rewritten. Chebyshev index link added. - Wolfdieter Lang, Aug 26 2014

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified September 24 22:31 EDT 2017. Contains 292441 sequences.