%I A004254 M3930
%S A004254 0,1,5,24,115,551,2640,12649,60605,290376,1391275,6665999,31938720,
%T A004254 153027601,733199285,3512968824,16831644835,80645255351,386394631920,
%U A004254 1851327904249,8870244889325,42499896542376,203629237822555
%N A004254 a(n) = 5a(n - 1) - a(n - 2), a(0) = 0, a(1) = 1.
%C A004254 Nonnegative values of y satisfying x^2 - 21*y^2 = 4; values of x are 
               in A003501. - W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               Nov 29 2002
%D A004254 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A004254 E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 
               7 (1969), pps. 231-242.
%D A004254 F. A. Haight, On a generalization of Pythagoras' theorem, pp. 73-77 of 
               J. C. Butcher, editor, A Spectrum of Mathematics. Auckland University 
               Press, 1971.
%D A004254 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=5, q=-1.
%D A004254 A. F. Horadam, Pell Identities, Fib. Quart., Vol. 9, No. 3, 1971, pps. 
               245-252.
%D A004254 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=7.
%D A004254 F. M. van Lamoen, Square wreaths around hexagons, Forum Geometricorum, 
               6 (2006) 311-325.
%H A004254 T. D. Noe, <a href="b004254.txt">Table of n, a(n) for n=0..200</a>
%H A004254 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A004254 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 A004254 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 A004254 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A004254 F. M. van Lamoen, <a href="http://forumgeom.fau.edu/FG2006volume6/FG200637index.html">
               Article in Forum Geometricorum</a>
%H A004254 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A004254 G.f.: x/(1-5*x+x^2). a(n)= S(2*n-1, sqrt(7))/sqrt(7) = S(n-1, 5); S(n, 
               x)=U(n, x/2), Chebyshev polynomials of 2nd kind, A049310.
%F A004254 (A003501)=sqrt{[21*(a(n))^2]+4}.
%F A004254 a(n)={[((5+sqrt(21))/2)^n]-[((5-sqrt(21))/2)^n]}/[sqrt(21)]. - Barry 
               E. Williams, Aug 29 2000
%F A004254 a(n)=sum{k=0..n-1, binomial(n+k, 2k+1)2^k} - Paul Barry (pbarry(AT)wit.ie), 
               Nov 30 2004
%F A004254 [A004253(n), a(n)] = [1,3; 1,4]^n * [1,0] - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Mar 19 2008
%e A004254 a(n+1)=sum{k=0..n, Gegenbauer_C(n-k,k+1,2)}. [From Paul Barry (pbarry(AT)wit.ie), 
               Apr 21 2009]
%p A004254 A004254:=1/(1-5*z+z**2); [S. Plouffe in his 1992 dissertation.]
%o A004254 (PARI) a(n)=if(n<0,0,subst(4*poltchebi(n+1)-10*poltchebi(n),x,5/2)/21)
%o A004254 (PARI) a(n)=if(n<0,0,imag((5+quadgen(84))^n)/2^(n-1))
%o A004254 sage: [lucas_number1(n,5,1) for n in range(27)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jun 25 2008
%Y A004254 Partial sums of A004253.
%Y A004254 Cf. A000027, A001906, A001353, A003501, A030221. a(n) = sqrt((A003501(n)^2 
               - 4)/21).
%Y A004254 First differences of a(n) are in A004253, partial sums in A089817.
%Y A004254 Cf. A004253.
%Y A004254 INVERT transformation yields A001109. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Sep 11 2008]
%Y A004254 Sequence in context: A140766 A026388 A057969 this_sequence A086347 A026707 
               A110190
%Y A004254 Adjacent sequences: A004251 A004252 A004253 this_sequence A004255 A004256 
               A004257
%K A004254 easy,nonn
%O A004254 0,3
%A A004254 N. J. A. Sloane (njas(AT)research.att.com).
%E A004254 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Aug 31 2000