cp's OEIS Frontend

A001077 Numerators of continued fraction convergents to sqrt(5).

%I A001077 M1934 N0764 #204 Jul 13 2025 19:38:26
%S A001077 1,2,9,38,161,682,2889,12238,51841,219602,930249,3940598,16692641,
%T A001077 70711162,299537289,1268860318,5374978561,22768774562,96450076809,
%U A001077 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)) = A075796(n+1), n >= 0, with T(n,x), resp. S(n,x), Chebyshev's polynomials of the first, resp. second kind. See A053120, resp. A049310.
%C A001077 From _Greg Dresden_, May 21 2023: (Start)
%C A001077 For n >= 2, 8*a(n) is the number of ways to tile this T-shaped figure of length n-1 with four colors of squares and one color of domino; shown here is the figure of length 5 (corresponding to n=6), and it has 8*a(6) = 23112 different tilings.
%C A001077     _
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%C A001077 (End)
%D A001077 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001077 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A001077 V. Thébault, Les Récréations Mathématiques, Gauthier-Villars, Paris, 1952, p. 282.
%H A001077 G. C. Greubel, <a href="/A001077/b001077.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..200 from T. D. Noe)
%H A001077 E. I. Emerson, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/7-3/emerson.pdf">Recurrent Sequences in the Equation DQ^2=R^2+N</a>, Fib. Quart., 7 (1969), pp. 231-242, Ex. 1, pp. 237-238.
%H A001077 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A001077 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A001077 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A001077 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A001077 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,1).
%F A001077 G.f.: (1-2*x)/(1-4*x-x^2).
%F A001077 a(n) = 4*a(n-1) + a(n-2), a(0)=1, a(1)=2.
%F A001077 a(n) = ((2 + sqrt(5))^n + (2 - sqrt(5))^n)/2.
%F A001077 a(n) = A014448(n)/2.
%F A001077 Limit_{n->infinity} a(n)/a(n-1) = phi^3 = 2 + sqrt(5). - _Gregory V. Richardson_, 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_, May 10 2003
%F A001077 E.g.f.: exp(2x)cosh(sqrt(5)x). - _Paul Barry_, May 10 2003
%F A001077 a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k)*5^k*2^(n-2k). - _Paul Barry_, 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_, Mar 19 2005
%F A001077 a(n) = F(3*n)/2 + F(3*n-1) where F() = Fibonacci numbers A000045. - _Gerald McGarvey_, Apr 28 2007
%F A001077 a(n) = A000032(3*n)/2.
%F A001077 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
%F A001077 a(n) = Sum_{k=0..n} A201730(n,k)*4^k. - _Philippe Deléham_, Dec 06 2011
%F A001077 a(n) = A001076(n) + A015448(n). - _R. J. Mathar_, Jul 06 2012
%F A001077 G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(5*k-4)/(x*(5*k+1) - 2/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 27 2013
%F A001077 a(n) is the (1,1)-entry of the matrix W^n with W=[2, sqrt(5); sqrt(5), 2]. - _Carmine Suriano_, Mar 21 2014
%F A001077 From _Rigoberto Florez_, Apr 03 2019: (Start)
%F A001077 a(n) = A099919(n) + A049651(n) if n > 0.
%F A001077 a(n) = 1 + Sum_{k=0..n-1} L(3*k + 1) if n >= 0, L(n) = n-th Lucas number (A000032). (End)
%F A001077 From _Christopher Hohl_, Aug 22 2021: (Start)
%F A001077 For n >= 2, a(2n-1) = A079962(6n-9) + A079962(6n-3).
%F A001077 For n >= 1, a(2n) = sqrt(20*A079962(6n-3)^2 + 1). (End)
%F A001077 a(n) = Sum_{k=0..n-2} A168561(n-2,k)*4^k + 2 * Sum_{k=0..n-1} A168561(n-1,k)*4^k, n>0. - _R. J. Mathar_, Feb 14 2024
%F A001077 a(n) = 4^n*Sum_{k=0..n} A374439(n, k)*(-1/4)^k. - _Peter Luschny_, Jul 26 2024
%F A001077 From _Peter Bala_, Jul 08 2025: (Start)
%F A001077 The following series telescope:
%F A001077 Sum_{n >= 1} 1/(a(n) + 5*(-1)^(n+1)/a(n)) = 3/8, since 1/(a(n) + 5*(-1)^(n+1)/a(n)) = b(n) - b(n+1), where b(n) = (1/4) * (a(n) + a(n-1)) / (a(n)*a(n-1)).
%F A001077 Sum_{n >= 1} (-1)^(n+1)/(a(n) + 5*(-1)^(n+1)/a(n)) = 1/8, since 1/(a(n) + 5*(-1)^(n+1)/a(n)) = c(n) + c(n+1), where c(n) = (1/4) * (a(n) - a(n-1)) / (a(n)*a(n-1)). (End)
%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 _Simon Plouffe_ in his 1992 dissertation
%p A001077 with(combinat): a:=n->fibonacci(n+1, 4)-2*fibonacci(n, 4): seq(a(n), n=0..30); # _Zerinvary Lajos_, Apr 04 2008
%t A001077 LinearRecurrence[{4, 1}, {1, 2}, 30]
%t A001077 Join[{1},Numerator[Convergents[Sqrt[5],30]]] (* _Harvey P. Dale_, Mar 23 2016 *)
%t A001077 CoefficientList[Series[(1-2*x)/(1-4*x-x^2), {x, 0, 30}], x] (* _G. C. Greubel_, Dec 19 2017 *)
%t A001077 LucasL[3*Range[0,30]]/2 (* _Rigoberto Florez_, Apr 03 2019 *)
%t A001077 a[ n_] := LucasL[n, 4]/2; (* _Michael Somos_, Nov 02 2021 *)
%o A001077 (Sage) [lucas_number2(n,4,-1)/2 for n in range(0, 30)] # _Zerinvary Lajos_, May 14 2009
%o A001077 (PARI) {a(n) = fibonacci(3*n) / 2 + fibonacci(3*n - 1)}; /* _Michael Somos_, Aug 11 2009 */
%o A001077 (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
%o A001077 (PARI) x='x+O('x^30); Vec((1-2*x)/(1-4*x-x^2)) \\ _G. C. Greubel_, Dec 19 2017
%o A001077 (Magma) I:=[1, 2]; [n le 2 select I[n] else 4*Self(n-1) + Self(n-2): n in [1..30]]; // _G. C. Greubel_, Dec 19 2017
%Y A001077 Cf. A000032, A001076, A023039, A049629, A052924, A078343, A164581, A179237, A180148, A329723, A374439.
%K A001077 nonn,easy,frac,nice
%O A001077 0,2
%A A001077 _N. J. A. Sloane_
%E A001077 Chebyshev comments from _Wolfdieter Lang_, Jan 10 2003