A295337 -id:A295337 - cp's OEIS frontend

A294969 Decimal expansion of sqrt(14)/2 = sqrt(7/2) = A010471/2.

1, 8, 7, 0, 8, 2, 8, 6, 9, 3, 3, 8, 6, 9, 7, 0, 6, 9, 2, 7, 9, 1, 8, 7, 4, 3, 6, 6, 1, 5, 8, 2, 7, 4, 6, 5, 0, 8, 7, 8, 0, 0, 9, 9, 0, 3, 8, 8, 9, 3, 6, 3, 4, 7, 3, 1, 5, 1, 8, 7, 2, 7, 3, 3, 6, 6, 0, 0, 1, 7, 5, 7, 8, 1, 5, 3, 4, 6, 9, 5, 1, 3, 9, 8, 8, 4, 0, 4, 9, 4, 7, 5, 9, 7, 1, 8, 9, 7, 8

Offset: 1

Wolfdieter Lang, Nov 27 2017

The regular continued fraction of sqrt(14)/2 is [1, repeat(1, 6, 1, 2)].
The convergents are given in A295336/A295337.
sqrt(14)/2 appears in a regular hexagon inscribed in a circle of radius 1 unit in the following way. Draw a straight line through two opposed midpoints of a side (halving the hexagon). The length between one of the midpoints, say M, and one of the two vertices nearest to the opposed midpoint is sqrt(13)/2 = A295330 units. A circle through M with this length ratio sqrt(13)/2 intersects the line below the hexagon at a point, say P. Then the length ratio between P and one of the two vertices nearest to M is sqrt(14)/2 (from a right triangle (1/2, sqrt(13)/2, sqrt(14)/2)).

			1.87082869338697069279187436615827465087800990388936347315187273366001757815...

Cf. A010471, A295330, A295336/A295337.

First[RealDigits[Sqrt[14]/2, 10, 100]] (* Paolo Xausa, May 23 2025 *)

A295336 Numerators of the convergents to sqrt(14)/2 = A294969.

Original entry on oeis.org

1, 2, 13, 15, 43, 58, 391, 449, 1289, 1738, 11717, 13455, 38627, 52082, 351119, 403201, 1157521, 1560722, 10521853, 12082575, 34687003, 46769578, 315304471, 362074049, 1039452569, 1401526618, 9448612277, 10850138895

Offset: 0

Wolfdieter Lang, Nov 27 2017

The corresponding denominators are given in A295337.
The recurrence is a(n) = b(n)*a(n-1) + a(n-2), n >= 1, with a(0) = 1, a(-1) = 1, with b(n) from the continued fraction b = {1,repeat(1, 6, 1, 2)}.
The g.f.s G_j(x) = Sum_{n>=0} a(4*n+j)*x^k, for j=1..4 satisfy (arguments are omitted): G_0 = 1 + 2*x*G_3 + x*G_2, G_1= G_0 + 1 + x*G_3, G_2 = 6*G_1 + G_0, G_3 = G_2 + G_1. After solving for the G_j(x), one finds for G(x) = Sum_{n>=0} a(n)*x^n = Sum_{j=1..4} x^j*G_j(x^4) the o.g.f. given in the formula section.

			The convergents a(n)/A295337(n) begin: 1, 2, 13/7, 15/8, 43/23, 58/31, 391/209, 449/240, 1289/689, 1738/929, 11717/6263, 13455/7192, 38627/20647, 52082/27839, 351119/187681, 403201/215520, 1157521/618721, 1560722/834241, ...

Index entries for linear recurrences with constant coefficients, signature (0,0,0,30,0,0,0,-1).

Cf. A294969, A295337.

Numerator[Convergents[Sqrt[14]/2, 50]] (* Vaclav Kotesovec, Nov 29 2017 *)
LinearRecurrence[{0,0,0,30,0,0,0,-1},{1,2,13,15,43,58,391,449},50] (* Harvey P. Dale, Apr 11 2022 *)

G.f.: (1 + 2*x + 13*x^2 + 15*x^3 + 13*x^4 - 2*x^5 + x^6 - x^7)/(1 - 30*x^4 + x^8).

a(n) = 30*a(n-4) - a(n-8), n >= 8, with inputs a(0)..a(7).

cp's OEIS Frontend

A294969 Decimal expansion of sqrt(14)/2 = sqrt(7/2) = A010471/2.

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