A041006 - OEIS

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

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OFFSET

0,1

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).

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

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 *)

PROG

(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

From M. F. Hasler, Nov 01 2019: (Start)

(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:

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} \\ (End)

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).

Sequence in context: A357020 A181443 A041165 * A346557 A288028 A083465

Adjacent sequences: A041003 A041004 A041005 * A041007 A041008 A041009

KEYWORD

nonn,cofr,frac,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Vincenzo Librandi, Jun 10 2015

STATUS

approved