A329592 -id:A329592 - cp's OEIS frontend

A329591 Decimal expansion of sqrt(34 + 2*sqrt(17))/4 = sqrt(8 + A222132)/2.

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

Offset: 1

Wolfdieter Lang, Feb 17 2020

The present cp := sqrt(34 + 2*sqrt(17))/4 is used, together with cm := sqrt(34 - 2*sqrt(17))/4 = sqrt(9 - A222132)/2 = A329592, for the roots of the integer polynomial P(4, x) := x^4 + x^3 - 6*x^2 - x + 1 which are x1 = 4 + cp - 2*cp^2, x2 = 4 - cp - 2*cp^2, x3 = 4 + cm - 2*cm^2, and x4 = 4 - cm - 2*cm^2. The approximate values of these zeros are 0.344150732, -2.905703544, 2.049481177, and -0.4879283650, respectively.

In the power basis of cp (denoted by (...)) and cm (denoted by [...]) the roots of P(4, x) are therefore: (4, +1, -2), (4, -1, -2), [4, +1, -2] and [4, -1, -2], respectively.

			1.62492713781332594517011169187886610389245001466924916684547590815419...

Cf, A222132, A329592.

RealDigits[Sqrt[34 + 2*Sqrt[17]]/4, 10, 100][[1]] (* Amiram Eldar, Feb 17 2020 *)

cp := sqrt(34 + 2*sqrt(17))/4 = sqrt(8 + w(17))/2, where w(17) = (1 - sqrt(17))/2 = A222132.

A370393 Decimal expansion of the area of a unit heptadecagon (17-gon).

Original entry on oeis.org

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

Offset: 2

Michal Paulovic, Feb 17 2024

This constant multiplied by the square of the side length of a regular heptadecagon equals the area of that heptadecagon.
17^2 divided by this constant equals 68 * tan(Pi/17) = 12.71140300... which is the perimeter and the area of an equable heptadecagon with its side length 4 * tan(Pi/17) = 0.74772958... .
An equable rectangle with its perimeter and area = 17 has side lengths:
  a = s^2/8 = (17 - sqrt(17)) / 4 = (17 - A010473) / 4 = 3.21922359...
  b = 136/s^2 = (17 + sqrt(17)) / 4 = (17 + A010473) / 4 = 5.28077640...
  where s is the parameter from the formula mentioned below.

			22.7354918984165514...

Eric Weisstein's World of Mathematics, Heptadecagon.
Wikipedia, Heptadecagon.

Cf. A007450, A010473, A019684 (Pi/17), A210644 (cos(2*Pi/17)), A210649, A228787, A241243, A329592, A343061.

Maple
```
evalf(17 / (4 * tan(Pi/17)), 100);
```

RealDigits[17 / (4 * Tan[Pi/17]), 10, 100][[1]]

PARI
```
17 / (4 * tan(Pi/17))
```

Equals 17 / (4 * tan(Pi/17)) = 17 / (4 * A343061).
Equals 1 / (4 * A007450 * A343061).
Equals 17 * cos(Pi/17) / (4 * sin(Pi/17)).
Equals 17 * A210649 / (4 * A241243).
Equals 17 * A210649 / (2 * A228787).
Equals 17 * cot(Pi/17) / 4.
Equals 17 * sqrt(4 / (s^2 - 2 * s - 4 * sqrt(17 + 3 * sqrt(17) - s - sqrt(17) * 16/s)) - 1/16) where s = sqrt(34 - 2 * sqrt(17)) = 4 * A329592.
The minimal polynomial is 4294967296*x^16 - 3103113871360*x^14 + 510054948143104*x^12 - 28954726431195136*x^10 + 653743432704327680*x^8 - 6011468019822067712*x^6 + 20881180982314634240*x^4 - 21552361799603318912*x^2 + 2862423051509815793.

cp's OEIS Frontend

A329591 Decimal expansion of sqrt(34 + 2*sqrt(17))/4 = sqrt(8 + A222132)/2.

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