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A078922 a(n) = 11*a(n-1) - a(n-2). 12
1, 10, 109, 1189, 12970, 141481, 1543321, 16835050, 183642229, 2003229469, 21851881930, 238367471761, 2600190307441, 28363725910090, 309400794703549, 3375045015828949, 36816094379414890, 401601993157734841 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

All positive integer solutions of Pell equation (3*b(n))^2 - 13*a(n)^2 = -4 together with b(n)=A097783(n-1), n>=1.

a(n) = L(n-1,11), where L is defined as in A108299; see also A097783 for L(n,-11). - Reinhard Zumkeller, Jun 01 2005

Number of 01-avoiding words of length n on alphabet {0,1,2,3,4,5,6,7,8,9, A} which do not end in 0. - Tanya Khovanova, Jan 10 2007

LINKS

Table of n, a(n) for n=1..18.

S. Falcon, Relationships between Some k-Fibonacci Sequences, Applied Mathematics, 2014, 5, 2226-2234.

Tanya Khovanova, Recursive Sequences

J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014

Index entries for sequences related to Chebyshev polynomials..

Index entries for linear recurrences with constant coefficients, signature (11,-1)

FORMULA

a(1)=1, a(2)=10 and for n>2 a(n) = ceiling(g*f^n) where f=(11+sqrt(117))/2 and g=(1-3/sqrt(13))/2 - Benoit Cloitre, Jan 12 2003

a(n)a(n+3) = 99 + a(n+1)a(n+2). - Ralf Stephan, May 29 2004

a(n) = S(n-1, 11) - S(n-2, 11) = T(2*n-1, sqrt(13)/2)/(sqrt(13)/2).

a(n+1) = ((-1)^n)*S(2*n, I*3), n>=0, with the imaginary unit I and S(n, x)=U(n, x/2) Chebyshev's polynomials of the second kind, A049310.

G.f.: x*(1-x)/(1-11*x+x^2).

a(n) = A006190(2*n-1). - Vladimir Reshetnikov, Sep 16 2016

EXAMPLE

All positive solutions of the Pell equation x^2 - 13*y^2 = -4 are

(x,y)= (3=3*1,1), (36=3*12,10), (393=3*131,109), (4287=3*1429,1189 ), ...

MATHEMATICA

LinearRecurrence[{11, -1}, {1, 10}, 20] (* Harvey P. Dale, Jan 26 2014 *)

Table[Fibonacci[2n - 1, 3], {n, 1, 20}] (* Vladimir Reshetnikov, Sep 16 2016 *)

PROG

(PARI) a(n)=([0, 1; -1, 11]^n*[1; 1])[1, 1] \\ Charles R Greathouse IV, Jun 11 2015

CROSSREFS

Row 11 of array A094954.

Cf. similar sequences listed in A238379.

Sequence in context: A198700 A267280 A015591 * A199760 A082181 A190919

Adjacent sequences:  A078919 A078920 A078921 * A078923 A078924 A078925

KEYWORD

nonn,easy

AUTHOR

Nick Renton (ner(AT)nickrenton.com), Jan 11 2003

EXTENSIONS

More terms from Benoit Cloitre, Jan 12 2003

STATUS

approved

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Last modified November 18 12:21 EST 2017. Contains 294891 sequences.