login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A041006 Numerators of continued fraction convergents to sqrt(6). 15

%I #45 Sep 08 2022 08:44:53

%S 2,5,22,49,218,485,2158,4801,21362,47525,211462,470449,2093258,

%T 4656965,20721118,46099201,205117922,456335045,2030458102,4517251249,

%U 20099463098,44716177445,198964172878,442644523201,1969542265682,4381729054565,19496458483942

%N Numerators of continued fraction convergents to sqrt(6).

%C Interspersion of 2 sequences, 2*A054320 and A001079. - _Gerry Martens_, Jun 10 2015

%H Hugo Pfoertner, <a href="/A041006/b041006.txt">Table of n, a(n) for n = 0..100</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,10,0,-1).

%F From _M. F. Hasler_, Feb 13 2009: (Start)

%F a(2n) = 2*A142238(2n) = A041038(2n)/2;

%F a(2n-1) = A142238(2n-1) = A041038(2n-1) = A001079(n). (End)

%F G.f.: (2 + 5*x + 2*x^2 - x^3)/(1 - 10*x^2 + x^4).

%t Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[6],n]]],{n,1,50}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 16 2011*)

%t LinearRecurrence[{0, 10, 0, -1}, {2, 5, 22, 49}, 50] (* _Vincenzo Librandi_, Jun 10 2015 *)

%o (Magma) I:=[2, 5, 22, 49]; [n le 4 select I[n] else 10*Self(n-2)-Self(n-4): n in [1..30]]; // _Vincenzo Librandi_, Jun 10 2015

%o From _M. F. Hasler_, Nov 01 2019: (Start)

%o (PARI) A41006=contfracpnqn(c=contfrac(sqrt(6)), #c)[1, ][^-1] \\ Discard possibly incorrect last element. NB: a(n)=A41006[n+1]! For correct index & more terms:

%o A041006(n)={n<#A041006|| A041006=extend(A041006, [2, 10; 4, -1], n\.8); A041006[n+1]}

%o extend(A, c, N)={for(n=#A+1, #A=Vec(A, N), A[n]=[A[n-i]|i<-c[, 1]]*c[, 2]); A} \\ (End)

%Y Cf. A041007 (denominators).

%Y Cf. A054320, A001079.

%Y Analog for other sqrt(m): A001333 (m=2), A002531 (m=3), A001077 (m=5), A041008 (m=7), A041010 (m=8), A005667 (m=10), A041014 (m=11), ..., A042936 (m=1000).

%K nonn,cofr,frac,easy

%O 0,1

%A _N. J. A. Sloane_

%E More terms from _Vincenzo Librandi_, Jun 10 2015

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)