%I A001076 M3538 N1434
%S A001076 0,1,4,17,72,305,1292,5473,23184,98209,416020,1762289,7465176,
%T A001076 31622993,133957148,567451585,2403763488,10182505537,43133785636,
%U A001076 182717648081,774004377960,3278735159921,13888945017644
%N A001076 Denominators of continued fraction convergents to sqrt(5).
%C A001076 a(2*n+1) with b(2*n+1) := A001077(2*n+1), n>=0, give all (positive integer) 
               solutions to Pell equation b^2 - 5*a^2 = -1, a(2*n) with b(2*n) := 
               A001077(2*n), n>=1, give all (positive integer) solutions to Pell 
               equation b^2 - 5*a^2 = +1 (cf. Emerson reference).
%C A001076 Bisection: a(2*n+1)= T(2*n+1,sqrt(5))/sqrt(5)= A007805(n), n>=0 and a(2*n)=4*S(n-1,
               18),n>=0, with T(n,x), resp. S(n,x), Chebyshev's polynomials of the 
               first,resp. second kind. S(-1,x)=0. See A053120, resp. A049310. S(n,
               18)=A049660(n+1).
%C A001076 Apart from initial terms, this is the Pisot sequence E(4,17), a(n)=[ 
               a(n-1)^2/a(n-2)+1/2 ].
%C A001076 This is also the Horadam sequence (0,1,1,4), having the recurrence relation 
               a(n) = s*a(n-1) + r*a(n-2); for n > 1, where a(0) = 0, a(1) = 1, 
               s = 4, r = 1. a(n) / a(n-1) converges to 5^1/2 + 2 as n approaches 
               infinity. 5^1/2 + 2 can also be written as (2 * Phi) + 1 and Phi^2 
               + Phi. - Ross La Haye (rlahaye(AT)new.rr.com), Aug 18 2003
%C A001076 Numerators in continued fraction [2, 4, 4, 4,...] = (1, 4, 17, 72,...) 
               = numerators of continued fraction [4, 4, 4,...]; where the convergents 
               to [4, 4, 4,...] = (1/4, 4/17, 17/72,...). Let X = the 2 X 2 matrix 
               [0, 1; 1, 4]; then X^n = [a(n-1), a(n); a(n), a(n+1)]; e.g. X^3 = 
               [4, 17; 17, 72]. Let C = the limit of a(n)/a(n-1) = 2 + sqrt(5) = 
               4.236067977...; then C^n = a(n+1) + (1/C)*a(n), where (1/C) = .236067977.... 
               Example: C^3 = 76.01315556..., = 72 + 17*(.2360679....). - Gary W. 
               Adamson (qntmpkt(AT)yahoo.com), Dec 15 2007
%C A001076 Sqrt(5) = 4/2 + 4/17 + 4/(17*305) + 4/(305*5473) + 4/(5473*98209) +... 
               - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 15 2007
%C A001076 a(p) == 20^((p-1)/2)) mod p, for odd primes p. [From Gary W. Adamson 
               (qntmpkt(AT)yahoo.com), Feb 22 2009]
%C A001076 A001076 == One halfs of even Fibonacci numbers. [From Vladimir Orlovsky 
               (4vladimir(AT)gmail.com), Oct 25 2009]
%C A001076 a(n) = A167808(3*n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Nov 12 2009]
%D A001076 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001076 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001076 S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques 
               Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%D A001076 D. W. Boyd, Some integer sequences related to the Pisot sequences, Acta 
               Arithmetica, 34 (1979), 295-305.
%D A001076 D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, 
               Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. 
               Publ., Oxford Univ. Press, New York, 1993.
%D A001076 E. I. Emerson, Recurrent sequences in the equation DQ^2=R^2+N, Fib. Quart., 
               7 (1969), 231-242, Thm. 1, p. 233.
%D A001076 V. Th\'{e}bault, Les R\'{e}cr\'{e}ations Math\'{e}matiques. Gauthier-Villars, 
               Paris, 1952, p. 282.
%D A001076 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of 
               combinatorial proof, M.A.A. 2003, id. 23.
%H A001076 T. D. Noe, <a href="b001076.txt">Table of n, a(n) for n=0..200</a>
%H A001076 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A001076 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 A001076 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 A001076 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A001076 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=398">
               Encyclopedia of Combinatorial Structures 398</a>
%H A001076 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A001076 a(n) = 4a(n-1) + a(n-2), n>1. a(0)=0, a(1)=1. G.f.: x/(1-4*x-x^2).
%F A001076 a(n)=((2+sqrt(5))^n - (2-sqrt(5))^n)/(2*sqrt(5)).
%F A001076 a(n) = ((-i)^(n-1))*S(n-1, 4*i), with i^2 =-1 and S(n, x) := U(n, x/2) 
               Chebyshev's polynomials of the second kind. See A049310. S(-1, x)=0.
%F A001076 a(n)=F(3n)/F(3), with F(n) Fibonacci numbers. - Mario Catalani (mario.catalani(AT)unito.it), 
               Jul 24 2003
%F A001076 a(n)=sum{i=0..n, sum{j=0..n, Fib(i+j)*n!/(i!j!(n-i-j)!)/2}} - Paul Barry 
               (pbarry(AT)wit.ie), Feb 06 2004
%F A001076 E.g.f.: exp(2*x)*sinh(sqrt(5)*x)/sqrt(5). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Sep 01 2004
%F A001076 a(n) = F(1) + F(4) + F(7) + ... + F(3n-2), for n>0.
%F A001076 Conjecture: 2a(n+1) = a(n+2) - A001077(n+1); Sequences (a(n)), A001077 
               generated by floretion: 'ii' + 'jj' - 'kk' + 0.5'ik' + 0.5'ki' - 
               e - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Nov 
               28 2004
%F A001076 a(n)=sum{k=0..n, sum{j=0..n, C(n, j)C(j, k)F(j)/2}} - Paul Barry (pbarry(AT)wit.ie), 
               Feb 14 2005
%F A001076 a(n) = A048876(n) - A048875(n) - Creighton Dement (crowdog(AT)t-online.de), 
               Mar 19 2005
%F A001076 Let M = {{0, 1}, {1, 4}}, v[1] = {0, 1}, v[n] = M.v[n - 1]; then a(n) 
               = v[n][[1]]. - Roger L. Bagula (rlbagulatftn(AT)yahoo.com), May 29 
               2005 - T. D. Noe (noe(AT)sspectra.com), Jan 19 2006
%F A001076 a(n)=F(n, 4), the n-th Fibonacci polynomial evaluated at x=4. - T. D. 
               Noe (noe(AT)sspectra.com), Jan 19 2006
%F A001076 [A015448(n), a(n)] = [1,4; 1,3]^n * [1,0] - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Mar 21 2008
%e A001076 1 2 9 38 161 (A001077)
%e A001076 -,-,-,--,---, ...
%e A001076 0 1 4 17 72 (A001076)
%e A001076 x + 4*x^2 + 17*x^3 + 72*x^4 + 305*x^5 + 1292*x^6 + 5473*x^7 + 23184*x^8 
               + ... - Michael Somos Aug 11 2009
%p A001076 K:=1/(1+4*z-z^2): Kser:=series(K, z=0, 30): seq(abs(coeff(Kser, z, n)), 
               n= -1..21); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 
               08 2007
%p A001076 A001076:=-1/(-1+4*z+z**2); [Conjectured by S. Plouffe in his 1992 dissertation.]
%p A001076 with(combinat): a:=n->fibonacci(n,4)-4*fibonacci(n-1,4): seq(a(n), n=2..24); 
               - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 04 2008
%t A001076 Clear[f,lst,n,a] f[n_]:=Fibonacci[n]; lst={};Do[a=f[n];If[EvenQ[a],AppendTo[lst,
               a/2]],{n,0,5!}];lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), 
               Oct 25 2009]
%t A001076 a=0;lst={a};s=0;Do[a=s-(a-1);AppendTo[lst,a];s+=a*4,{n,3*4!}];lst [From 
               Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 27 2009]
%o A001076 (Mupad) numlib::fibonacci(3*n)/2 $ n = 0..30; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               May 09 2008
%o A001076 sage: from sage.combinat.sloane_functions import recur_gen3 sage: it 
               = recur_gen3(0,1,4,4,1,0) sage: [it.next() for i in xrange(1,32)] 
               - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 09 2008
%o A001076 (Other) sage: [lucas_number1(n,4,-1) for n in xrange(0, 23)]# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Apr 23 2009]
%o A001076 (Other) sage: [fibonacci(3*n)/2 for n in xrange(0, 23)]# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), May 15 2009]
%o A001076 (PARI) {a(n) = fibonacci(3*n) / 2} - Michael Somos Aug 11 2009
%Y A001076 Cf. A001077, A015448, A033887.
%Y A001076 A001076(n)=F(3n)/2, where F=A000045 (the Fibonacci sequence).
%Y A001076 Cf. A049660, A007805.
%Y A001076 Partial sums of A033887. First differences of A049652. Bisection of A059973.
%Y A001076 Third column of array A028412.
%Y A001076 Cf. A015448.
%Y A001076 A014445 [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 15 2009]
%Y A001076 Sequence in context: A136792 A108929 A022031 this_sequence A122451 A113442 
               A085732
%Y A001076 Adjacent sequences: A001073 A001074 A001075 this_sequence A001077 A001078 
               A001079
%K A001076 nonn,easy,cofr,nice,new
%O A001076 0,3
%A A001076 N. J. A. Sloane (njas(AT)research.att.com).
%E A001076 Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               Jan 10 2003