Search: id:A001077
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%I A001077 M1934 N0764
%S A001077 1,2,9,38,161,682,2889,12238,51841,219602,930249,3940598,
%T A001077 16692641,70711162,299537289,1268860318,5374978561,22768774562,
%U A001077 96450076809,408569081798,1730726404001,7331474697802,31056625195209
%N A001077 Numerators of continued fraction convergents to sqrt(5).
%C A001077 a(2*n+1) with b(2*n+1) := A001076(2*n+1), n>=0, give all (positive integer) 
               solutions to Pell equation a^2 - 5*b^2 = -1.
%C A001077 a(2*n) with b(2*n) := A001076(2*n), n>=1, give all (positive integer) 
               solutions to Pell equation a^2 - 5*b^2 = +1 (see Emerson reference).
%C A001077 Bisection: a(2*n)= T(n,9)= A023039(n), n>=0 and a(2*n+1)=2*S(2*n,2*sqrt(5)),
               n>=0, with T(n,x), resp. S(n,x), Chebyshev's polynomials of the first,
               resp. second kind. See A053120, resp. A049310.
%D A001077 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001077 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001077 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 A001077 E. I. Emerson, Recurrent sequences in the equation DQ^2=R^2+N, Fib. Quart., 
               7 (1969), 231-242, Ex.1, p. 237-8.
%D A001077 V. Th\'{e}bault, Les R\'{e}cr\'{e}ations Math\'{e}matiques. Gauthier-Villars, 
               Paris, 1952, p. 282.
%H A001077 T. D. Noe, <a href="b001077.txt">Table of n, a(n) for n=0..200</a>
%H A001077 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A001077 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 A001077 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 A001077 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A001077 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A001077 G.f.: (1-2*x)/(1-4*x-x^2); a(n)=4*a(n-1)+a(n-2), a(0)=1, a(1)=2; a(n)=[ 
               (2+sqrt(5))^n + (2-sqrt(5))^n ]/2.
%F A001077 Lim. n-> Inf. a(n)/a(n-1) = phi^3 = 2 + Sqrt(5). - Gregory V. Richardson 
               (omomom(AT)hotmail.com), Oct 13 2002
%F A001077 a(n) = ((-i)^n)*T(n, 2*i), with T(n, x) Chebyshev's polynomials of the 
               first kind A053120 and i^2 = -1.
%F A001077 Binomial transform of A084057. - Paul Barry (pbarry(AT)wit.ie), May 10 
               2003
%F A001077 E.g.f.: exp(2x)cosh(sqrt(5)x) - Paul Barry (pbarry(AT)wit.ie), May 10 
               2003
%F A001077 a(n)=sum{k=0..floor(n/2), C(n, 2k)5^k2^(n-2k)} - Paul Barry (pbarry(AT)wit.ie), 
               Nov 15 2003
%F A001077 a(n) = 4*a(n-1) + a(n-2) when n > 2; a[1] = 1, a[2] = 2. - Alex Vinokur 
               (alexvn(AT)barak-online.net), Oct 25 2004
%F A001077 a(n) = A001076(n+1) - 2*A001076(n) = A097924(n) - A015448(n+1); a(n+1) 
               = A097924(n) + 2*A001076(n) = A097924(n) + 2(A048876(n) - A048875(n)) 
               - Creighton Dement (crowdog(AT)t-online.de), Mar 19 2005
%F A001077 a(n) = F(3*n)/2 + F(3*n-1) where F() = Fibonacci numbers A000045. - Gerald 
               McGarvey (gerald.mcgarvey(AT)comcast.net), Apr 28 2007
%e A001077 1 2 9 38 161 (A001077)
%e A001077 -,-,-,--,---, ...
%e A001077 0 1 4 17 72 (A001076)
%e A001077 1 + 2*x + 9*x^2 + 38*x^3 + 161*x^4 + 682*x^5 + 2889*x^6 + 12238*x^7 + 
               ... - Michael Somos Aug 11 2009
%p A001077 A001077:=(-1+2*z)/(-1+4*z+z**2); [Conjectured by S. Plouffe in his 1992 
               dissertation.]
%p A001077 with(combinat): a:=n->fibonacci(n,4)-2*fibonacci(n-1,4): seq(a(n), n=1..23); 
               - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 04 2008
%o A001077 (Other) sage: [lucas_number2(n,4,-1)/2 for n in xrange(0, 23)]# [From 
               Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 14 2009]
%o A001077 (PARI) {a(n) = fibonacci(3*n) / 2 + fibonacci(3*n - 1)} - Michael Somos 
               Aug 11 2009
%Y A001077 A001077(n)=A014448(n)/2.
%Y A001077 Cf. A001076.
%Y A001077 Cf. A023039, A049629.
%Y A001077 Sequence in context: A007224 A037489 A037569 this_sequence A150993 A150994 
               A150995
%Y A001077 Adjacent sequences: A001074 A001075 A001076 this_sequence A001078 A001079 
               A001080
%K A001077 nonn,easy,cofr,nice
%O A001077 0,2
%A A001077 N. J. A. Sloane (njas(AT)research.att.com).
%E A001077 Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               Jan 10 2003

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