%I A001090 M4554 N1936
%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
%N A001090 a(n) = 8*a(n-1)-a(n-2); a(0) = 0, a(1) = 1.
%C A001090 a(n)=((4+Sqrt(15))^n - (4-Sqrt(15))^n))/(2*Sqrt(15)) [From Sture Sjoestedt 
               (sture.sjostedt(AT)spray.se), May 31 2009]
%C A001090 Number of units of a(n) belongs to a periodic sequence: 0, 1, 8, 3, 6, 
               5, 4, 7, 2, 9. [From Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 
               04 2009]
%D A001090 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001090 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001090 H. Brocard, Notes e'le'mentaires sur le proble`me de Peel, Nouvelles 
               Correspondance Math\'{e}matique, 4 (1878), 161-169.
%D A001090 E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 
               7 (1969), pps. 231-242.
%D A001090 A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. 
               Quart., 5.5 (1967), 424-434. Case a=0,b=1; p=8, q=-1.
%D A001090 W. Lang, On polynomials related to powers of the generating function 
               of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eq.(44), lhs, 
               m=10.
%H A001090 T. D. Noe, <a href="b001090.txt">Table of n, a(n) for n=0..100</a>
%H A001090 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A001090 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A001090 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A001090 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A001090 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A001090 15*a(n)^2 - A001091(n)^2 = -1.
%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 a(n)={{(4+sqrt(15))^n} - {(4-sqrt(15))^n}}/2*sqrt(15). G.f.(x)=x/(1-8x+x^2). 
               - Barry E. Williams, Aug 18 2000
%F A001090 Lim. n-> Inf. a(n)/a(n-1) = 4 + sqrt(15). - Gregory V. Richardson (omomom(AT)hotmail.com), 
               Oct 13 2002
%F A001090 a(n) = 7*(a(n-1)+a(n-2))-a(n-3). a(n) = 9*(a(n-1)-a(n-2))+a(n-3). - Mohamed 
               Bouhamida (bhmd95(AT)yahoo.fr), Feb 07 2007
%F A001090 [A070997(n-1), a(n)] = [1,6; 1,7]^n * [1,0]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Mar 21 2008
%F A001090 a(-n) = -a(n). - Michael Somos Apr 05 2008
%F A001090 a(n+1)=8a(n) - a(n-1) a(1)=1 , a(2)=8 [From Sture Sjoestedt (sture.sjostedt(AT)spray.se), 
               May 31 2009]
%e A001090 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); [S. Plouffe in his 1992 dissertation.]
%t A001090 lst={};Do[AppendTo[lst, GegenbauerC[n, 1, 4]], {n, 0, 6^2}];lst [From 
               Vladimir Orlovsky (4vladimir(AT)gmail.com), Sep 11 2008]
%o A001090 (PARI) {a(n) = subst(poltchebi(n+1) - 4 * poltchebi(n), x, 4) / 15} /
               * Michael Somos Apr 05 2008 */
%o A001090 sage: [lucas_number1(n,8,1) for n in range(22)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jun 25 2008
%o A001090 (Other) sage: [lucas_number1(n,8,1) for n in xrange(0, 21)]# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Apr 23 2009]
%Y A001090 Cf. A000027, A001906, A001353, A004254, A001109, A004187, A001091.
%Y A001090 a(n)=sqrt((A001091(n)^2-1)/15).
%Y A001090 Cf. A070997.
%Y A001090 Sequence in context: A085433 A081107 A164592 this_sequence A105219 A037205 
               A060071
%Y A001090 Adjacent sequences: A001087 A001088 A001089 this_sequence A001091 A001092 
               A001093
%K A001090 nonn
%O A001090 0,3
%A A001090 N. J. A. Sloane (njas(AT)research.att.com).
%E A001090 More terms from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               Aug 02 2000