A295331 -id:A295331 - cp's OEIS frontend

A295330 Decimal expansion of sqrt(13)/2.

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

Offset: 1

Wolfdieter Lang, Nov 20 2017

In a regular hexagon inscribed in a circle of radius R the largest distance between any vertex and a midpoint of a side, after division of R, is sqrt(13)/2. The two smaller distance ratios are sqrt(7)/2 = A242703 and 1/2.
The regular period 6 continued fraction of sqrt(13)/2 is [1; 1, 4, 14, 4, 1, 2], and the convergents are given in A295331/A295332.
Essentially the same as A223139, A209927, A098316 and A085550. - R. J. Mathar, Nov 23 2017

			1.8027756377319946465596106337352479731256482869226231063552265281135834741465...

Index entries for algebraic numbers, degree 2.

Cf. A010470, A242703, A295331/A295332.

RealDigits[Sqrt@13/2, 10, 111][[1]] (* Robert G. Wilson v, Nov 20 2017 *)

sqrt(13)/2 \\ Michel Marcus, Nov 21 2017

A295332 Denominators of the continued fraction convergents to sqrt(13)/2 = A295330.

Original entry on oeis.org

1, 1, 5, 71, 289, 360, 1009, 1369, 6485, 92159, 375121, 467280, 1309681, 1776961, 8417525, 119622311, 486906769, 606529080, 1699964929, 2306494009, 10925940965, 155269667519, 632004611041, 787274278560, 2206553168161, 2993827446721, 14181862955045, 201539908817351, 820341498224449, 1021881407041800, 2864104312308049, 3885985719349849, 18408047189707445

Offset: 0

Wolfdieter Lang, Nov 20 2017

The numerators are given in A295331.

The continued fraction expansion of sqrt(13)/2 is [1, repeat(1, 4, 14, 4, 1, 2)].

			See A295331 for the first convergents.

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 1298, 0, 0, 0, 0, 0, -1).

Cf. A295330, A295331.

G.f.: (1 + x + 5*x^2 + 71*x^3 + 289*x^4 + 360*x^5 - 289*x^6 + 71*x^7 - 5*x^8 + x^9 - x^10) / ((1 - 3*x - x^2)*(1 + 3*x - x^2)*(1 + 3*x + 10*x^2 - 3*x^3 + x^4)*(1 - 3*x + 10*x^2 + 3*x^3 + x^4)). See A295331 for a hint for the derivation. Here the a(n) recurrence is the same as there but the inputs are a(0) = 1, a(-1) = 0, (a(-2) = 1). The unfactorized denominator is 1 - 1298*x^6 + x^12.

a(n) = 1298*a(n-6) - a(n-12), n >= 12, with inputs a(0)..a(11).

cp's OEIS Frontend

A295330 Decimal expansion of sqrt(13)/2.

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