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A157014 Expansion of x*(1-x)/(x^2-22*x+1). 16
1, 21, 461, 10121, 222201, 4878301, 107100421, 2351330961, 51622180721, 1133336644901, 24881784007101, 546265911511321, 11992968269241961, 263299036011811821, 5780585823990618101, 126909589091781786401, 2786230374195208682721, 61170158643202809233461 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

This sequence is part of a solution of a general problem involving 2 equations, three sequences a(n), b(n), c(n) and a constant A:

    A    * c(n)+1 = a(n)^2,

   (A+1) * c(n)+1 = b(n)^2, where solutions are given by the recurrences:

a(1) = 1, a(2) = 4*A+1, a(n) = (4*A+2)*a(n-1)-a(n-2) for n>2, resulting in a(n) terms 1, 4*A+1, 16*A^2+12*A+1, 64*A^3+80*A^2+24*A+1, ...;

b(1) = 1, b(2) = 4*A+3, b(n) = (4*A+2)*b(n-1)-b(n-2) for n>2, resulting in b(n) terms 1, 4*A+3, 16*A^2+20*A+5, 64*A^3+112*A^2+56*A+7, ...;

c(1) = 0, c(2) = 16*A+8, c(3) = (16*A^2+16*A+3)*c(2), c(n) = (16*A^2+16*A+3) * (c(n-1)-c(n-2)) + c(n-3) for n>3, resulting in c(n) terms 0, 16*A+8, 256*A^3+384*A^2+176*A+24, 4096*A^5 + 10240*A^4 + 9472*A^3 + 3968*A^2 + 736*A + 48, ... .

A157014 is the a(n) sequence for A=5.

For other A values the a(n), b(n) and c(n) sequences are in the OEIS:

A   a-sequence    b-sequence     c-sequence

1     A001653       A002315(n-1)   A078522

2     A072256       A054320(n-1)   A045502(n-1)

3     A001570       A028230        A059989(n-1)

4     A007805       A049629(n-1)   A157459

5  -> A157014 <-    A133283        A157460

6     A153111       A157461        A157874

7     A157877       A157878        A157879

8     A077420(n-1)  A046176        A157880

9     A097315(n-1)  A097314(n-1)   A157881

Positive values of x (or y) satisfying x^2 - 22xy + y^2 + 20 = 0. - Colin Barker, Feb 19 2014

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Index entries for linear recurrences with constant coefficients, signature (22, -1).

FORMULA

G.f.: x*(1-x)/(x^2-22*x+1).

a(1) = 1, a(2) = 21, a(n) = 22*a(n-1)-a(n-2) for n>2.

5*A157460(n)+1 = a(n)^2 for n>=1.

6*A157460(n)+1 = A133283(n)^2 for n>=1.

a(n) = (6+sqrt(30)-(-6+sqrt(30))*(11+2*sqrt(30))^(2*n))/(12*(11+2*sqrt(30))^n). - Gerry Martens, Jun 07 2015

MATHEMATICA

CoefficientList[Series[(1 - x)/(x^2 - 22 x + 1), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 21 2014 *)

a[c_, n_] := Module[{},

   p := Length[ContinuedFraction[ Sqrt[ c]][[2]]];

   d := Denominator[Convergents[Sqrt[c], n p]];

   t := Table[d[[1 + i]], {i, 0, Length[d] - 1, p}];

   Return[t];

] (* Complement of A041049 *)

a[30, 20] (* Gerry Martens, Jun 07 2015 *)

PROG

(PARI) Vec((1-x)/(x^2-22*x+1)+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

(MAGMA) I:=[1, 21, 461]; [n le 2 select I[n] else 22*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 21 2014

CROSSREFS

Cf. similar sequences listed in A238379.

Sequence in context: A199252 A199197 A219419 * A076552 A126996 A158603

Adjacent sequences:  A157011 A157012 A157013 * A157015 A157016 A157017

KEYWORD

nonn,easy

AUTHOR

Paul Weisenhorn, Feb 21 2009

EXTENSIONS

Edited by Alois P. Heinz, Sep 09 2011

STATUS

approved

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Last modified October 17 01:24 EDT 2018. Contains 316275 sequences. (Running on oeis4.)