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

2, 5, 22, 49, 218, 485, 2158, 4801, 21362, 47525, 211462, 470449, 2093258, 4656965, 20721118, 46099201, 205117922, 456335045, 2030458102, 4517251249, 20099463098, 44716177445, 198964172878, 442644523201, 1969542265682, 4381729054565, 19496458483942

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• frac
• easy

Comments

Interspersion of 2 sequences, 2*A054320 and A001079. - Gerry Martens, Jun 10 2015

Links

• Hugo Pfoertner, Table of n, a(n) for n = 0..100
• Index entries for linear recurrences with constant coefficients, signature (0,10,0,-1).

Crossrefs

Cf. A041007 (denominators).

Cf. A054320, A001079.

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).

Programs

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

Mathematica

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[6],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2011 *)
LinearRecurrence[{0, 10, 0, -1}, {2, 5, 22, 49}, 50] (* Vincenzo Librandi, Jun 10 2015 *)

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

PARI

\\ For correct index & more terms:
A041006(n)={n<#A041006|| A041006=extend(A041006, [2, 10; 4, -1], n\.8); A041006[n+1]}
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

Formula

From M. F. Hasler, Feb 13 2009: (Start)

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

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

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

a(n) = ((2 + sqrt(6))^(n+1) + (2 - sqrt(6))^(n+1))/2^(ceiling(n/2) + 1). - Robert FERREOL, Oct 13 2024

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

Extensions

More terms from Vincenzo Librandi, Jun 10 2015