A331115 Numerators of the best approximations for sqrt(2).

1, 3, 4, 7, 17, 24, 41, 99, 140, 239, 577, 816, 1393, 3363, 4756, 8119, 19601, 27720, 47321, 114243, 161564, 275807, 665857, 941664, 1607521, 3880899, 5488420, 9369319, 22619537, 31988856, 54608393, 131836323, 186444716, 318281039, 768398401, 1086679440, 1855077841, 4478554083, 6333631924

Offset: 1

Author

Gerhard Kirchner, Jan 10 2020

Keywords

• nonn
• frac
• easy

Comments

Every principal convergent, see A002965, and every second intermediate convergent is a best approximation for sqrt(2). The numerators of these convergents are the terms of the current sequence. For denominators and more details, see A331101.

Examples

			The principal convergents are 1/1, 3/2, 7/5, 17/12, ... and 1,3,7,17,... the corresponding numerators, see A001333. Intermediate convergents: (2/1), 4/3, (10/7), 24/17, ... (best approximations without brackets). Numerators: 4,24,... (subsequence of A143607). All these numerators sorted: 1,3,4,7,17,24,...
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Crossrefs

Cf. A002965, A331101, A001333, A143607, A000129.

Programs

Mathematica

LinearRecurrence[{0,0,6,0,0,-1},{1,3,4,7,17,24},50] (* Harvey P. Dale, Nov 26 2024 *)

PARI

Vec(x*(1 + x)*(1 + 2*x + 2*x^2 - x^3) / (1 - 6*x^3 + x^6) + O(x^40)) \\ Colin Barker, Jan 10 2020

Formula

If n mod 3 = 2: a(n) = 3*a(n-1) - a(n-2), otherwise: a(n) = a(n-1) + a(n-2), for n>2 with a(1)=1, a(2)=3.

a(3n-2) = 1/2*S(2n-1), a(3n-1) = 1/2*S(2n), a(3n) = w/2*D(2n), for n>0 with w = sqrt(2) and S(n) = (1+w)^n + (1-w)^n and D(n) = (1+w)^n - (1-w)^n.

From Colin Barker, Jan 10 2020: (Start)

G.f.: x*(1 + x)*(1 + 2*x + 2*x^2 - x^3) / (1 - 6*x^3 + x^6).

a(n) = 6*a(n-3) - a(n-6) for n>6.

(End)