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

Author

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

Keywords

• nonn
• easy

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

Examples

			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 ), ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..960
• R. C. Alperin, A family of nonlinear recurrences and their linear solutions, Fib. Q., 57:4 (2019), 318-321.
• Sergio 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).

Crossrefs

Row 11 of array A094954.

Cf. similar sequences listed in A238379.

Programs

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

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( x*(1-x)/(1-11*x+x^2) )); // G. C. Greubel, Jan 12 2019

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 *)

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

Sage

(x*(1-x)/(1-11*x+x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 12 2019

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

Extensions

More terms from Benoit Cloitre, Jan 12 2003

Definition adapted to offset by Georg Fischer, Jun 18 2021