The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A078922 a(n) = 11*a(n-1) - a(n-2) with a(1)=1, a(2) = 10. 13
 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 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:=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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified August 10 05:56 EDT 2024. Contains 375044 sequences. (Running on oeis4.)