A294973 -id:A294973 - cp's OEIS frontend

A242703 Decimal expansion of sqrt(7)/2.

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

Offset: 1

Alonso del Arte, May 20 2014

Absolute value of the imaginary part of any of the nontrivial divisors of 2 in O_Q(sqrt(-7)).
The incircle of a triangle with sides of lengths 4, 5, 6 units respectively has a radius of sqrt(7)/2.
With a different offset, decimal expansion of 5 * sqrt(7).
From Wolfdieter Lang, Nov 18 2017: (Start)
In a regular hexagon inscribed in a circle with a radius of 1 unit the three distinct distances between any vertex and the middle of the sides are 1/2, sqrt(7)/2 and sqrt(13)/2.
The continued fraction expansion of sqrt(7)/2 is 1, repeat(3, 10, 3, 2). The convergents are given in A294972/A294973. (End)

			1.32287565553229529525...

Iain Fox, Table of n, a(n) for n = 1..20000 (first 1000 terms from Ivan Panchenko)

Cf. A010527, A020837, A040022, A294972/A294973.

Mathematica
```
RealDigits[Sqrt[7]/2, 10, 100][[1]]
```

{ default(realprecision, 20080); x=sqrt(7)/2; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b242703.txt", n, " ", d)); } \\ Iain Fox, Nov 18 2017

(1/2 - sqrt(-7)/2)(1/2 + sqrt(-7)/2) = 2.

Equals A010465/2. - R. J. Mathar, Feb 23 2021

A294972 Numerators of continued fraction convergents to sqrt(7)/2.

Original entry on oeis.org

1, 4, 41, 127, 295, 1012, 10415, 32257, 74929, 257044, 2645369, 8193151, 19031671, 65288164, 671913311, 2081028097, 4833969505, 16582936612, 170663335625, 528572943487, 1227809222599, 4212000611284, 43347815335439, 134255446617601, 311858708570641, 1069831572329524, 11010174431865881

Offset: 0

Wolfdieter Lang, Nov 18 2017

The denominators are given in A294973.

The continued fraction expansion of sqrt(7)/2 is 1, repeat(3, 10, 3, 2).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,254,0,0,0,-1).

Cf. A242703, A294973.

Numerator[Convergents[Sqrt[7]/2, 30]] (* Vaclav Kotesovec, Nov 19 2017 *)

Vec((1 + 4*x + 41*x^2 + 127*x^3 + 41*x^4 - 4*x^5 + x^6 - x^7) / ((1 - 16*x^2 + x^4)*(1 + 16*x^2 + x^4)) + O(x^40)) \\ Colin Barker, Nov 21 2017

From Colin Barker, Nov 19 2017: (Start)
G.f.: (1 + 4*x + 41*x^2 + 127*x^3 + 41*x^4 - 4*x^5 + x^6 - x^7) / ((1 - 16*x^2 + x^4)*(1 + 16*x^2 + x^4)).
a(n) = 254*a(n-4) - a(n-8) for n > 7.
(End)
The proof of the g.f. runs like the one given for the denominators in A294973. The recurrence for a(n) is the same but the input is now a(0) = b(0) = 1 and a(-1) = 1, (a(-2) = 0). - Wolfdieter Lang, Nov 19 2017

A295331 Numerators of continued fraction convergents to sqrt(13)/2 = A295330.

Original entry on oeis.org

1, 2, 9, 128, 521, 649, 1819, 2468, 11691, 166142, 676259, 842401, 2361061, 3203462, 15174909, 215652188, 877783661, 1093435849, 3064655359, 4158091208, 19697020191, 279916373882, 1139362515719, 1419278889601, 3977920294921, 5397199184522, 25566717033009, 363331237646648, 1478891667619601, 1842222905266249, 5163337478152099, 7005560383418348, 33185579011825491

Offset: 0

Wolfdieter Lang, Nov 20 2017

The denominators are given in A295332.

The continued fraction expansion of sqrt(13)/2 is [1, repeat(1, 4, 14, 4, 1, 2)] = [b(0), repeat(b(1), b(2), b(3), b(4), b(5), b(6))].

			The convergents a(n)/A295332 begin: 1, 2, 9/5, 128/71, 521/289, 649/360, 1819/1009, 2468/1369, 11691/6485, 166142/92159, 676259/375121, 842401/467280, 2361061/1309681, 3203462/1776961, 15174909/8417525, 215652188/119622311, 877783661/486906769, 1093435849/606529080, ...

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 1298, 0, 0, 0, 0, 0, -1).

Cf. A294972/A294973 (sqrt(7/2)), A295330, A295332.

Numerator[Convergents[Sqrt[13]/2,40]] (* or *) LinearRecurrence[ {0,0,0,0,0,1298,0,0,0,0,0,-1},{1,2,9,128,521,649,1819,2468,11691,166142,676259,842401},40] (* Harvey P. Dale, Jan 23 2019 *)

G.f.: (1 + 2*x + 9*x^2 + 128*x^3 + 521*x^4 + 649*x^5 + 521*x^6 - 128*x^7 + 9*x^8 + 2*x^9 + x^10 - x^11) / ((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)). For the derivation see the period 4 example for the denominators of the convergents of sqrt(7)/2 given in A294972. Here the period is 6 and the input for the recurrence a(n) = b(n)*a(n-1) + a(n-2) is a(-1) = 1 = a(0) (a(-2) = 1 - b(0) = 0) with the b(n) modulo 6 given above. Note that a(6*k) = 2*a(6*k-1) + a(6*k-2) is valid only for k >= 1 because a(0) = 1 is not equal to 2*a(-1) + a(-2) = 2. The denominator in the unfactorized form 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

A242703 Decimal expansion of sqrt(7)/2.

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A294972 Numerators of continued fraction convergents to sqrt(7)/2.

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A295331 Numerators of continued fraction convergents to sqrt(13)/2 = A295330.

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