login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A266703 Coefficient of x^2 in minimal polynomial of the continued fraction [1^n,2/3,1,1,1,...], where 1^n means n ones. 3
9, 11, 1, 29, 45, 149, 359, 971, 2511, 6605, 17261, 45221, 118359, 309899, 811295, 2124029, 5560749, 14558261, 38113991, 99783755, 261237231, 683927981, 1790546669, 4687712069, 12272589495, 32130056459, 84117579839, 220222683101, 576550469421, 1509428725205 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
See A265762 for a guide to related sequences.
LINKS
FORMULA
a(n) = 2*a(n-1) - 2*a(n-2) + a(n-3).
G.f.: (9 - 7 x - 39 x^2 + 14 x^3 - 4 x^4 + 2 x^5)/(1 - 2 x - 2 x^2 + x^3).
a(n) = 2^(-n)*(-43*(-2)^n+(3+sqrt(5))^n*(-1+3*sqrt(5))-(3-sqrt(5))^n*(1+3*sqrt(5)))/5 for n>2. - Colin Barker, Sep 29 2016
EXAMPLE
Let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:
[2/3,1,1,1,...] = (1+3*sqrt(5))/6 has p(0,x) = -11 - 3 x + 9 x^2, so a(0) = 9;
[1,2/3,1,1,...] = (19+9*sqrt(5))/22 has p(1,x) = -1 - 19 x + 11 x^2, so a(1) = 11;
[1,1,2/3,1,...] = (-17+9*sqrt(5))/2 has p(2,x) = -29 + 17 x + x^2, so a(2) = 1.
MATHEMATICA
u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {2/3}, {{1}}];
f[n_] := FromContinuedFraction[t[n]];
t = Table[MinimalPolynomial[f[n], x], {n, 0, 20}]
Coefficient[t, x, 0] (* A266703 *)
Coefficient[t, x, 1] (* A266704 *)
Coefficient[t, x, 2] (* A266703 *)
PROG
(PARI) Vec((9-7*x-39*x^2+14*x^3-4*x^4+2*x^5)/((1+x)*(1-3*x+x^2)) + O(x^30)) \\ Colin Barker, Sep 29 2016
CROSSREFS
Sequence in context: A119207 A165254 A058069 * A266701 A172283 A172185
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jan 09 2016
STATUS
approved

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

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 21 15:14 EDT 2024. Contains 374474 sequences. (Running on oeis4.)