cp's OEIS Frontend

A041011 Denominators of continued fraction convergents to sqrt(8).

%I A041011 #89 Feb 16 2025 08:32:38
%S A041011 1,1,5,6,29,35,169,204,985,1189,5741,6930,33461,40391,195025,235416,
%T A041011 1136689,1372105,6625109,7997214,38613965,46611179,225058681,
%U A041011 271669860,1311738121,1583407981,7645370045,9228778026,44560482149
%N A041011 Denominators of continued fraction convergents to sqrt(8).
%C A041011 Sqrt(8) = 2 + continued fraction [0; 1, 4, 1, 4, 1, 4, ...] = 4/2 + 4/5 + 4/(5*29) + 4/(29*169) + 4/(169*985) + ... - _Gary W. Adamson_, Dec 21 2007
%C A041011 This is the sequence of Lehmer numbers U_n(sqrt(R),Q) with the parameters R = 4 and Q = -1. It is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for all natural numbers n and m. - _Peter Bala_, May 12 2014
%C A041011 Apparently the same as A152118(n). - _Georg Fischer_, Jul 01 2019
%H A041011 Vincenzo Librandi, <a href="/A041011/b041011.txt">Table of n, a(n) for n = 0..199</a> [shifted by _Georg Fischer_, Jul 01 2019]
%H A041011 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Barry4/barry64.html">Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays</a>, JIS 12 (2009) 09.8.6.
%H A041011 Hongshen Chua, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Chua2/chua13.html">A Study of Second-Order Linear Recurrence Sequences via Continuants</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.8.8.
%H A041011 J. L. Ramirez and F. Sirvent, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Ramirez/ramirez9.html">A q-Analogue of the Bi-Periodic Fibonacci Sequence</a>, J. Int. Seq. 19 (2016) # 16.4.6, t_n at a=1, b=4.
%H A041011 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LehmerNumber.html">Lehmer Number</a>
%H A041011 <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H A041011 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,6,0,-1).
%F A041011 a(2n) = A000129(2n+1), a(2n+1) = A000129(2n+2)/2.
%F A041011 a(n) = 6*a(n-2) - a(n-4). Also:
%F A041011 a(2n) = a(2n-1)+a(2n-2), a(2n+1)=4*a(2n)+a(2n-1).
%F A041011 G.f.: (1+x-x^2)/(1-6*x^2+x^4).
%F A041011 From _Peter Bala_, May 12 2014: (Start)
%F A041011 For n even, a(n) = (alpha^n - beta^n)/(alpha - beta), and for n odd, a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2), where alpha = 1 + sqrt(2) and beta = 1 - sqrt(2).
%F A041011 a(n) = Product_{k = 1..floor((n-1)/2)} ( 4 + 4*cos^2(k*Pi/n) ) for n >= 1. (End)
%F A041011 From _Gerry Martens_, Jul 11 2015: (Start)
%F A041011 Interspersion of 2 sequences [a0(n),a1(n)] for n>0:
%F A041011 a0(n) = ((3-2*sqrt(2))^n*(2+sqrt(2))-(-2+sqrt(2))*(3+2*sqrt(2))^n)/4.
%F A041011 a1(n) = (-(3-2*sqrt(2))^n+(3+2*sqrt(2))^n)/(4*sqrt(2)). (End)
%F A041011 a(n) = ((-(-1 - sqrt(2))^n - 3*(1-sqrt(2))^n + (-1+sqrt(2))^n + 3*(1+sqrt(2))^n))/(8*sqrt(2)). - _Colin Barker_, Mar 27 2016
%p A041011 with(combinat): a := n -> fibonacci(n + 1, 2)/2^(n mod 2):
%p A041011 seq(a(n), n = 0 .. 28); # _Miles Wilson_, Aug 04 2024
%t A041011 Denominator[NestList[(4/(4 + #))&, 0, 60]] (* _Vladimir Joseph Stephan Orlovsky_, Apr 13 2010 *)
%t A041011 CoefficientList[Series[(x + x^2 - x^3)/(1 - 6 x^2 + x^4), {x, 0, 40}], x] (* _Vincenzo Librandi_, Dec 10 2013 *)
%t A041011 a0[n_] := ((3-2*Sqrt[2])^n*(2+Sqrt[2])-(-2+Sqrt[2])*(3+2*Sqrt[2])^n)/4 // Simplify
%t A041011 a1[n_] := (-(3-2*Sqrt[2])^n+(3+2*Sqrt[2])^n)/(4*Sqrt[2]) // Simplify
%t A041011 Flatten[MapIndexed[{a0[#], a1[#]} &,Range[20]]] (* _Gerry Martens_, Jul 11 2015 *)
%t A041011 LinearRecurrence[{0,6,0,-1},{1,1,5,6},40] (* _Harvey P. Dale_, Oct 21 2024 *)
%o A041011 (Magma) I:=[1, 1, 5, 6]; [n le 4 select I[n] else 6*Self(n-2)-Self(n-4): n in [1..30]]; // _Vincenzo Librandi_, Dec 10 2013
%o A041011 (PARI) a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,0,6,0]^n*[1;1;5;6])[1,1] \\ _Charles R Greathouse IV_, Nov 13 2015
%o A041011 (PARI) my(x='x+O('x^99)); concat(0, Vec((1+x-x^2)/(1-6*x^2+x^4))) \\ _Altug Alkan_, Mar 27 2016
%Y A041011 Cf. A000129, A010466, A041010. A089499, A136211, A152118.
%K A041011 nonn,cofr,frac,easy
%O A041011 0,3
%A A041011 _N. J. A. Sloane_
%E A041011 Entry improved by _Michael Somos_
%E A041011 First term 0 in b-file, formulas and programs removed by _Georg Fischer_, Jul 01 2019