cp's OEIS Frontend

A041006 Numerators of continued fraction convergents to sqrt(6).

%I A041006 #64 Dec 29 2024 14:08:24
%S A041006 2,5,22,49,218,485,2158,4801,21362,47525,211462,470449,2093258,
%T A041006 4656965,20721118,46099201,205117922,456335045,2030458102,4517251249,
%U A041006 20099463098,44716177445,198964172878,442644523201,1969542265682,4381729054565,19496458483942
%N A041006 Numerators of continued fraction convergents to sqrt(6).
%C A041006 Interspersion of 2 sequences, 2*A054320 and A001079. - _Gerry Martens_, Jun 10 2015
%H A041006 Hugo Pfoertner, <a href="/A041006/b041006.txt">Table of n, a(n) for n = 0..100</a>
%H A041006 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,10,0,-1).
%F A041006 From _M. F. Hasler_, Feb 13 2009: (Start)
%F A041006 a(2n) = 2*A142238(2n) = A041038(2n)/2;
%F A041006 a(2n-1) = A142238(2n-1) = A041038(2n-1) = A001079(n). (End)
%F A041006 G.f.: (2 + 5*x + 2*x^2 - x^3)/(1 - 10*x^2 + x^4).
%F A041006 a(n) = ((2 + sqrt(6))^(n+1) + (2 - sqrt(6))^(n+1))/2^(ceiling(n/2) + 1). - _Robert FERREOL_, Oct 13 2024
%F A041006 E.g.f.: sqrt(2)*sinh(sqrt(2)*x)*(cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)) + cosh(sqrt(2)*x)*(2*cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)). - _Stefano Spezia_, Oct 14 2024
%t A041006 Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[6],n]]],{n,1,50}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 16 2011 *)
%t A041006 LinearRecurrence[{0, 10, 0, -1}, {2, 5, 22, 49}, 50] (* _Vincenzo Librandi_, Jun 10 2015 *)
%o A041006 (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 A041006 (PARI) A41006=contfracpnqn(c=contfrac(sqrt(6)), #c)[1, ][^-1] \\ Discard possibly incorrect last element. NB: a(n)=A41006[n+1]! _M. F. Hasler_, Nov 01 2019
%o A041006 (PARI) \\ For correct index & more terms:
%o A041006 A041006(n)={n<#A041006|| A041006=extend(A041006, [2, 10; 4, -1], n\.8); A041006[n+1]}
%o A041006 extend(A, c, N)={for(n=#A+1, #A=Vec(A, N), A[n]=[A[n-i]|i<-c[, 1]]*c[, 2]); A} \\ _M. F. Hasler_, Nov 01 2019
%Y A041006 Cf. A041007 (denominators).
%Y A041006 Cf. A054320, A001079.
%Y A041006 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 A041006 nonn,cofr,frac,easy
%O A041006 0,1
%A A041006 _N. J. A. Sloane_
%E A041006 More terms from _Vincenzo Librandi_, Jun 10 2015