# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/

Search: id:a001090
Showing 1-1 of 1

%I A001090 M4554 N1936 #171 Aug 17 2024 14:22:37
%S A001090 0,1,8,63,496,3905,30744,242047,1905632,15003009,118118440,929944511,
%T A001090 7321437648,57641556673,453811015736,3572846569215,28128961537984,
%U A001090 221458845734657,1743541804339272,13726875588979519,108071462907496880,850844827670995521,6698687158460467288
%N A001090 a(n) = 8*a(n-1) - a(n-2); a(0) = 0, a(1) = 1.
%C A001090 This sequence gives the values of y in solutions of the Diophantine equation x^2 - 15*y^2 = 1; the corresponding values of x are in A001091. - _Vincenzo Librandi_, Nov 12 2010 [edited by _Jon E. Schoenfield_, May 02 2014]
%C A001090 For n >= 2, a(n) equals the permanent of the (n-1) X (n-1) tridiagonal matrix with 8's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - _John M. Campbell_, Jul 08 2011
%C A001090 For n >= 1, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,7}. - _Milan Janjic_, Jan 25 2015
%C A001090 From _Klaus Purath_, Jul 25 2024: (Start)
%C A001090 For any three consecutive terms (x, y, z) y^2 - x*z = 1 always applies.
%C A001090 a(n) = (t(i+2n) - t(i))/(t(i+n+1) - t(i+n-1)) where (t) is any recurrence t(k) = 9t(k-1) - 9t(k-2) + t(k-3) or t(k) = 8t(k-1) - t(k-2) without regard to initial values.
%C A001090 In particular, if the recurrence (t) of the form (9,-9,1) has the initial values t(0) = 1, t(1) = 2, t(2) = 9, a(n) = t(n) - 1 applies. (End)
%D A001090 Julio R. Bastida, Quadratic properties of a linearly recurrent sequence. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 163--166, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561042 (81e:10009) - From _N. J. A. Sloane_, May 30 2012
%D A001090 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001090 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001090 G. C. Greubel, <a href="/A001090/b001090.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..100 from T. D. Noe)
%H A001090 Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, <a href="https://www.emis.de/journals/INTEGERS/papers/p38/p38.Abstract.html">Polynomial sequences on quadratic curves</a>, Integers, Vol. 15, 2015, #A38.
%H A001090 K. Andersen, L. Carbone, and D. Penta, <a href="https://pdfs.semanticscholar.org/8f0c/c3e68d388185129a56ed73b5d21224659300.pdf">Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields</a>, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.
%H A001090 H. Brocard, <a href="http://resolver.sub.uni-goettingen.de/purl?PPN598948236_0004/DMDLOG_0053">Notes élémentaires sur le problème de Peel</a>, Nouvelle Correspondance Mathématique, 4 (1878), 337-343.
%H A001090 E. I. Emerson, <a href="http://www.fq.math.ca/Scanned/7-3/emerson.pdf">Recurrent Sequences in the Equation DQ^2=R^2+N</a>, Fib. Quart., 7 (1969), pp. 231-242.
%H A001090 A. F. Horadam, <a href="http://www.fq.math.ca/Scanned/5-5/horadam.pdf">Special properties of the sequence W_n(a,b; p,q)</a>, Fib. Quart., 5.5 (1967), 424-434. Case a=0,b=1; p=8, q=-1.
%H A001090 Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Janjic/janjic63.html">On Linear Recurrence Equations Arising from Compositions of Positive Integers</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
%H A001090 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A001090 Wolfdieter Lang, <a href="http://www.fq.math.ca/Scanned/38-5/lang.pdf">On polynomials related to powers of the generating function of Catalan's numbers</a>, Fib. Quart. 38,5 (2000) 408-419; Eq.(44), lhs, m=10.
%H A001090 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A001090 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H A001090 Shobhna Singh and Felix Flicker, <a href="https://arxiv.org/abs/2309.14447">Exact Solution to the Quantum and Classical Dimer Models on the Spectre Aperiodic Monotiling</a>, arXiv:2309.14447 [cond-mat.str-el], 2023.
%H A001090 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A001090 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-1).
%F A001090 15*a(n)^2 - A001091(n)^2 = -1.
%F A001090 a(n) = sqrt((A001091(n)^2 - 1)/15).
%F A001090 a(n) = S(2*n-1, sqrt(10))/sqrt(10) = S(n-1, 8); S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310, with S(-1, x) := 0.
%F A001090 From _Barry E. Williams_, Aug 18 2000: (Start)
%F A001090 a(n) = ((4+sqrt(15))^n - (4-sqrt(15))^n)/(2*sqrt(15)).
%F A001090 G.f.: x/(1-8*x+x^2). (End)
%F A001090 Limit_{n->infinity} a(n)/a(n-1) = 4 + sqrt(15). - _Gregory V. Richardson_, Oct 13 2002
%F A001090 [A070997(n-1), a(n)] = [1,6; 1,7]^n * [1,0]. - _Gary W. Adamson_, Mar 21 2008
%F A001090 a(-n) = -a(n). - _Michael Somos_, Apr 05 2008
%F A001090 a(n+1) = Sum_{k=0..n} A101950(n,k)*7^k. - _Philippe Deléham_, Feb 10 2012
%F A001090 From _Peter Bala_, Dec 23 2012: (Start)
%F A001090 Product_{n >= 1} (1 + 1/a(n)) = (1/3)*(3 + sqrt(15)).
%F A001090 Product_{n >= 2} (1 - 1/a(n)) = (1/8)*(3 + sqrt(15)).
%F A001090 (End)
%F A001090 a(n) = A136325(n)/3. - _Greg Dresden_, Sep 12 2019
%F A001090 E.g.f.: exp(4*x)*sinh(sqrt(15)*x)/sqrt(15). - _Stefano Spezia_, Dec 12 2022
%F A001090 a(n) = Sum_{k = 0..n-1} binomial(n+k, 2*k+1)*6^k = Sum_{k = 0..n-1} (-1)^(n+k+1)* binomial(n+k, 2*k+1)*10^k. - _Peter Bala_, Jul 17 2023
%e A001090 G.f. = x + 8*x^2 + 63*x^3 + 496*x^4 + 3905*x^5 + 30744*x^6 + 242047*x^7 + ...
%p A001090 A001090:=1/(1-8*z+z**2); # _Simon Plouffe_ in his 1992 dissertation
%p A001090 seq( simplify(ChebyshevU(n-1, 4)), n=0..20); # _G. C. Greubel_, Dec 23 2019
%t A001090 Table[GegenbauerC[n-1, 1, 4]], {n,0,20}] (* _Vladimir Joseph Stephan Orlovsky_, Sep 11 2008 *)
%t A001090 LinearRecurrence[{8,-1},{0,1},30] (* _Harvey P. Dale_, Aug 29 2012 *)
%t A001090 a[n_]:= ChebyshevU[n-1, 4]; (* _Michael Somos_, May 28 2014 *)
%t A001090 CoefficientList[Series[x/(1-8*x+x^2), {x,0,20}], x] (* _G. C. Greubel_, Dec 20 2017 *)
%o A001090 (PARI) {a(n) = subst(poltchebi(n+1) - 4 * poltchebi(n), x, 4) / 15}; /* _Michael Somos_, Apr 05 2008 */
%o A001090 (PARI) {a(n) = polchebyshev(n-1, 2, 4)}; /* _Michael Somos_, May 28 2014 */
%o A001090 (PARI) my(x='x+O('x^30)); concat([0], Vec(x/(1-8*x-x^2))) \\ _G. C. Greubel_, Dec 20 2017
%o A001090 (SageMath) [lucas_number1(n,8,1) for n in range(22)] # _Zerinvary Lajos_, Jun 25 2008
%o A001090 (SageMath) [chebyshev_U(n-1,4) for n in (0..20)] # _G. C. Greubel_, Dec 23 2019
%o A001090 (Magma) I:=[0,1]; [n le 2 select I[n] else 8*Self(n-1) - Self(n-2): n in [1..30]]; // _G. C. Greubel_, Dec 20 2017
%o A001090 (GAP) m:=4;; a:=[0,1];; for n in [3..20] do a[n]:=2*m*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Dec 23 2019
%Y A001090 Cf. A001091, A001906, A004187, A004254, A049310, A070997.
%Y A001090 Equals one-third A136325.
%Y A001090 Chebyshev sequence U(n, m): A000027 (m=1), A001353 (m=2), A001109 (m=3), this sequence (m=4), A004189 (m=5), A004191 (m=6), A007655 (m=7), A077412 (m=8), A049660 (m=9), A075843 (m=10), A077421 (m=11), A077423 (m=12), A097309 (m=13), A097311 (m=14), A097313 (m=15), A029548 (m=16), A029547 (m=17), A144128 (m=18), A078987 (m=19), A097316 (m=33).
%Y A001090 Cf. A323182.
%K A001090 nonn,easy
%O A001090 0,3
%A A001090 _N. J. A. Sloane_
%E A001090 More terms from _Wolfdieter Lang_, Aug 02 2000

# Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE