A295333 Numerators of continued fraction convergents to sqrt(10)/2 = sqrt(5/2) = A020797 + 1.

1, 2, 3, 8, 11, 19, 49, 68, 117, 302, 419, 721, 1861, 2582, 4443, 11468, 15911, 27379, 70669, 98048, 168717, 435482, 604199, 1039681, 2683561, 3723242, 6406803, 16536848, 22943651, 39480499, 101904649, 141385148, 243289797, 627964742, 871254539, 1499219281, 3869693101, 5368912382, 9238605483

Offset: 0

Author

Wolfdieter Lang, Nov 21 2017

Keywords

• nonn
• frac
• cofr
• easy

Comments

The denominators are given in A295334.

The regular continued fraction expansion of sqrt(10)/2 is [1, repeat(1, 1, 2)].

Examples

			The convergents a(n)/A295334(n) begin: 1, 2, 3/2, 8/5, 11/7, 19/12, 49/31, 68/43, 117/74, 302/191, 419/265, 721/456, 1861/1177, 2582/1633, 4443/2810, 11468/7253, 15911/10063, 27379/17316, 70669/44695, 98048/62011, ...
		

Links

• Robert Israel, Table of n, a(n) for n = 0..3794
• Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,1).

Crossrefs

Cf. A020797, A295334.

Programs

Maple

numtheory:-cfrac(sqrt(5/2),100,'con'):
map(numer,con[1..-2]); # Robert Israel, Nov 22 2017

Mathematica

Numerator[Convergents[Sqrt[5/2], 50]] (* Vaclav Kotesovec, Nov 22 2017 *)
LinearRecurrence[{0,0,6,0,0,1},{1,2,3,8,11,19},40] (* Harvey P. Dale, Apr 08 2019 *)

Formula

G.f.: G(x) = (1 + 2*x + 3*x^2 + 2*x^3 - x^4 + x^5)/(1 - 6*x^3 - x^6). From the recurrence a(n) = b(n)*a(n-1) + a(n-2), with the trisection b(3*(k+1)) = 2, b(3*k+1) = 1 = b(3*k+2), k >= 0, b(0) = 1, and the input a(0) = 1 = a(-1). With G_j(x) = Sum_{k>=0} a(3*k+j)*x^k, for j = 0,1,2, one finds (omitting here the G_j arguments) G_0 = 1 + 2*x*G_2 + x*G_1, G_1 = G_0 + 1 + x*G_2, G_2 = G_1 + G_0. This can be solved and leads to the given formula for G(x) = Sum_{j=0..2} x^j*G_j(x^3).

a(n) = 6*a(n-3) + a(n-6), for n >= 6, with inputs a(0)..a(5).