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A078922 a(n) = 11*a(n-1) - a(n-2) with a(1)=1, a(2) = 10. 13

%I #68 Apr 20 2023 13:19:34

%S 1,10,109,1189,12970,141481,1543321,16835050,183642229,2003229469,

%T 21851881930,238367471761,2600190307441,28363725910090,

%U 309400794703549,3375045015828949,36816094379414890,401601993157734841

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

%C 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.

%C 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

%C 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

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

%H G. C. Greubel, <a href="/A078922/b078922.txt">Table of n, a(n) for n = 1..960</a>

%H S. Falcon, <a href="http://dx.doi.org/10.4236/am.2014.515216">Relationships between Some k-Fibonacci Sequences</a>, Applied Mathematics, 2014, 5, 2226-2234.

%H Alex Fink, Richard K. Guy, and Mark Krusemeyer, <a href="https://doi.org/10.11575/cdm.v3i2.61940">Partitions with parts occurring at most thrice</a>, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114. See Section 13.

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H Giovanni Lucca, <a href="http://forumgeom.fau.edu/FG2019volume19/FG201902index.html">Integer Sequences and Circle Chains Inside a Hyperbola</a>, Forum Geometricorum (2019) Vol. 19, 11-16.

%H J.-C. Novelli and J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv preprint arXiv:1403.5962 [math.CO], 2014.

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials..</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (11,-1).

%F 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

%F a(n)*a(n+3) = 99 + a(n+1)*a(n+2). - _Ralf Stephan_, May 29 2004

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

%F 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.

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

%F a(n) = A006190(2*n-1). - _Vladimir Reshetnikov_, Sep 16 2016

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

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

%t LinearRecurrence[{11,-1},{1,10},20] (* _Harvey P. Dale_, Jan 26 2014 *)

%t Table[Fibonacci[2n-1, 3], {n, 1, 20}] (* _Vladimir Reshetnikov_, Sep 16 2016 *)

%o (PARI) a(n)=([0,1;-1,11]^n*[1;1])[1,1] \\ _Charles R Greathouse IV_, Jun 11 2015

%o (PARI) my(x='x+O('x^30)); Vec(x*(1-x)/(1-11*x+x^2)) \\ _G. C. Greubel_, Jan 12 2019

%o (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( x*(1-x)/(1-11*x+x^2) )); // _G. C. Greubel_, Jan 12 2019

%o (Sage) (x*(1-x)/(1-11*x+x^2)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jan 12 2019

%o (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

%Y Row 11 of array A094954.

%Y Cf. similar sequences listed in A238379.

%K nonn,easy

%O 1,2

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

%E More terms from _Benoit Cloitre_, Jan 12 2003

%E Definition adapted to offset by _Georg Fischer_, Jun 18 2021

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Last modified March 28 07:18 EDT 2024. Contains 371235 sequences. (Running on oeis4.)