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

1,2

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.

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).

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)

Example

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,...

Prog

(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

Crossrefs

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

Sequence in context: A287493 A078825 A361659 * A327318 A344783 A093611

Adjacent sequences: A331112 A331113 A331114 * A331116 A331117 A331118

Keyword

nonn,frac,easy

Author

Gerhard Kirchner, Jan 10 2020

Status

approved