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A001077
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Numerators of continued fraction convergents to sqrt(5).
(Formerly M1934 N0764)
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32
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1, 2, 9, 38, 161, 682, 2889, 12238, 51841, 219602, 930249, 3940598, 16692641, 70711162, 299537289, 1268860318, 5374978561, 22768774562, 96450076809, 408569081798, 1730726404001, 7331474697802, 31056625195209
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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.
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).
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.
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REFERENCES
| E. I. Emerson, Recurrent sequences in the equation DQ^2=R^2+N, Fib. Quart., 7 (1969), 231-242, Ex.1, p. 237-8.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
V. Th\'{e}bault, Les R\'{e}cr\'{e}ations Math\'{e}matiques. Gauthier-Villars, Paris, 1952, p. 282.
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LINKS
| T. D. Noe, Table of n, a(n) for n = 0..200
Index entries for sequences related to linear recurrences with constant coefficients
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.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
| 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.
Lim. n-> Inf. a(n)/a(n-1) = phi^3 = 2 + Sqrt(5). - Gregory V. Richardson (omomom(AT)hotmail.com), Oct 13 2002
a(n) = ((-i)^n)*T(n, 2*i), with T(n, x) Chebyshev's polynomials of the first kind A053120 and i^2 = -1.
Binomial transform of A084057. - Paul Barry (pbarry(AT)wit.ie), May 10 2003
E.g.f.: exp(2x)cosh(sqrt(5)x) - Paul Barry (pbarry(AT)wit.ie), May 10 2003
a(n)=sum{k=0..floor(n/2), C(n, 2k)5^k2^(n-2k)} - Paul Barry (pbarry(AT)wit.ie), Nov 15 2003
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
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
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
a(n) = A000032(3*n)/2.
For n>=1: a(n) = 1/2*Fibonacci(6*n)/Fibonacci(3*n) and a(n) = integer part of (2+sqrt(5))^n. [Artur Jasinski, Nov 28 2011]
a(n) = Sum_{k, 0<=k<=n} A201730(n,k)*4^k . - DELEHAM Philippe, Dec 06 2011
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EXAMPLE
| 1 2 9 38 161 (A001077)
-,-,-,--,---, ...
0 1 4 17 72 (A001076)
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
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MAPLE
| A001077:=(-1+2*z)/(-1+4*z+z**2); [Conjectured by S. Plouffe in his 1992 dissertation.]
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
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MATHEMATICA
| LinearRecurrence[{4, 1}, {1, 2}, 30]
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PROG
| (Other) sage: [lucas_number2(n, 4, -1)/2 for n in xrange(0, 23)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 14 2009]
(PARI) {a(n) = fibonacci(3*n) / 2 + fibonacci(3*n - 1)} - Michael Somos Aug 11 2009
(PARI) a(n)=if(n<2, n+1, my(t=4); for(i=1, n-2, t=4+1/t); numerator(2+1/t)) \\ Charles R Greathouse IV, Dec 05 2011
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CROSSREFS
| A001077(n)=A014448(n)/2.
Cf. A001076, A023039, A049629, A000032 (Lucas Numbers).
Sequence in context: A181339 A037489 A037569 * A150993 A150994 A150995
Adjacent sequences: A001074 A001075 A001076 * A001078 A001079 A001080
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KEYWORD
| nonn,easy,frac,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jan 10 2003
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