A280725 -id:A280725 - cp's OEIS frontend

A280533 Decimal expansion of 14*sin(Pi/14).

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

Offset: 1

Rick L. Shepherd, Jan 04 2017

Decimal expansion of the ratio of the perimeter of a regular 7-gon (heptagon) to its diameter (largest diagonal).

			3.115293075388401660044635902955126632528977962703629374367818222563897248...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for algebraic numbers, degree 3.

Cf. For other n-gons: A010466 (n=4), 10*A019827 (n=5, 10), A280585 (n=8), A280633 (n=9), A280725 (n=11), A280819 (n=12).

Cf. A232736.

RealDigits[14*Sin[Pi/14], 10, 129][[1]] (* G. C. Greubel, Sep 20 2022 *)

PARI
```
14*sin(Pi/14)
```

numerical_approx(14*sin(pi/14), digits=127) # G. C. Greubel, Sep 20 2022

Equals 14*A232736.

A280585 Decimal expansion of 8*sin(Pi/8).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Jan 05 2017

Decimal expansion of the ratio of the perimeter of a regular 8-gon (octagon) to its diameter (largest diagonal).

			3.061467458920718173827679872243190934090756499885016331470405085020368271...

Index entries for algebraic numbers, degree 4.

Cf. For other n-gons: A010466 (n=4), 10*A019827 (n=5, 10), A280533 (n=7), A280633 (n=9), A280725 (n=11), A280819 (n=12).

Cf. A182168.

evalf(8*sin(Pi/8),100); # Wesley Ivan Hurt, Feb 01 2017

RealDigits[8*Sin[Pi/8], 10, 120][[1]] (* Amiram Eldar, Jun 26 2023 *)

PARI
```
8*sin(Pi/8)
```

Equals 8*A182168.

A280633 Decimal expansion of 18*sin(Pi/18).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Jan 06 2017

The ratio of the perimeter of a regular 9-gon (nonagon) to its diameter (largest diagonal).

Also least positive root of x^3 - 243x + 729.

			3.125667198004746279330899281847666328006762189313248970252344806377184798...

Index entries for algebraic numbers, degree 3.

Cf. For other n-gons: A010466 (n=4), 10*A019827 (n=5, 10), A280533 (n=7),A280585 (n=8), A280725(n=11), A280819 (n=12).

evalf(18*sin(Pi/18),100); # Wesley Ivan Hurt, Feb 01 2017

RealDigits[18*Sin[Pi/18],10,120][[1]] (* Harvey P. Dale, Dec 02 2018 *)

PARI
```
18*sin(Pi/18)
```

18*A019819.

A280819 Decimal expansion of 12*sin(Pi/12).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Jan 08 2017

The ratio of the perimeter of a regular 12-gon (dodecagon) to its diameter (greatest diagonal).

A quartic integer: the least positive root of x^4 - 144x^2 + 1296.

			3.105828541230249148186786051488579940188826815839166165768038487780683696...

Index entries for algebraic numbers, degree 4

Cf. For other n-gons: A010466 (n=4), 10*A019827 (n=5, 10), A280533 (n=7), A280585 (n=8), A280633 (n=9), A280725 (n=11).

First@ RealDigits@ N[12 Sin[Pi/12], 120] (* Michael De Vlieger, Feb 05 2017 *)

PARI
```
12*sin(Pi/12)
```

polrootsreal(x^4-144*x^2+1296)[3] \\ Charles R Greathouse IV, Feb 05 2017

12*A019824.

A284726 a(n) = (1/4) * smallest multiple of 4 missing from [A280864(1), ..., A280864(n-1)].

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 20, 20, 20, 21, 21, 21, 22, 22

Offset: 1

N. J. A. Sloane, Apr 06 2017

nonn

For k >= 1, n >= 1, let B_k(n) = smallest multiple of k missing from [A280864(1), ..., A280864(n-1)]. Sequence gives values of B_4(n)/4.

The analogous sequences B_k(n) for the EKG sequence A064413 were important for the analysis of that sequence, so they may also be useful for studying A280864.

			The initial terms of A280864 are 1,2,4,3,6,8,... The smallest missing multiple of 3 in [1,2,4,3,6] is 8, so a(6) = 8/4 = 2.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
J. C. Lagarias, E. M. Rains and N. J. A. Sloane, The EKG sequence, arXiv:math/0204011 [math.NT], 2002.
J. C. Lagarias, E. M. Rains and N. J. A. Sloane, The EKG Sequence, Exper. Math. 11 (2002), 437-446.

Cf. A280864, A064413, A284724, A280725.

mex := proc(L)
local k;
for k from 1 do
if not k in L then
return k;
end if;
end do:
end proc:
read b280864;
k:=4; a:=[1,1]; ML:=[]; B:=1;
for n from 2 to 120 do
t:=b280864[n];
if (t mod k) = 0 then
ML:=[op(ML),t/k];
B:=mex(ML);
a:=[op(a),B];
else
a:=[op(a),B];
fi;
od:
a;

terms = 84; rad[n_] := Times @@ FactorInteger[n][[All, 1]];
A280864 = Reap[present = 0; p = 1; pp = 1; Do[forbidden = GCD[p, pp]; mandatory = p/forbidden; a = mandatory; While[BitGet[present, a] > 0 || GCD[forbidden, a] > 1, a += mandatory]; Sow[a]; present += 2^a; pp = p; p = rad[a], terms]][[2, 1]];
Clear[a]; a[1] = 1;
a[n_] := a[n] = For[b = 4 a[n - 1], True, b += 4, If[FreeQ[A280864[[1 ;; n - 1]], b], Return[b/4]]];
Array[a, terms] (* Jean-François Alcover, Nov 26 2017, after Rémy Sigrist's program for A280864 *)

cp's OEIS Frontend

A280533 Decimal expansion of 14*sin(Pi/14).

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A280585 Decimal expansion of 8*sin(Pi/8).

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A280633 Decimal expansion of 18*sin(Pi/18).

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A280819 Decimal expansion of 12*sin(Pi/12).

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A284726 a(n) = (1/4) * smallest multiple of 4 missing from [A280864(1), ..., A280864(n-1)].

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Maple

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