A078922 a(n) = 11*a(n-1) - a(n-2) with a(1)=1, a(2) = 10.

1, 10, 109, 1189, 12970, 141481, 1543321, 16835050, 183642229, 2003229469, 21851881930, 238367471761, 2600190307441, 28363725910090, 309400794703549, 3375045015828949, 36816094379414890, 401601993157734841

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

References

R. C. Alperin, A family of nonlinear recurrences and their linear solutions, Fib. Q., 57:4 (2019), 318-321.

Links

G. C. Greubel, Table of n, a(n) for n = 1..960

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

Alex Fink, Richard K. Guy, and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114. See Section 13.

Tanya Khovanova, Recursive Sequences

Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16.

J.-C. Novelli and 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
(PARI) my(x='x+O('x^30)); Vec(x*(1-x)/(1-11*x+x^2)) \\ G. C. Greubel, Jan 12 2019
(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( x*(1-x)/(1-11*x+x^2) )); // G. C. Greubel, Jan 12 2019
(Sage) (x*(1-x)/(1-11*x+x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 12 2019
(GAP) a:=[1, 10];; for n in [3..30] do a[n]:=11*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 12 2019

Crossrefs

Row 11 of array A094954.

Cf. similar sequences listed in A238379.

Sequence in context: A320094 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

Definition adapted to offset by Georg Fischer, Jun 18 2021

Status

approved