cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A248355 Least k such that Pi - k*sin(Pi/k) < 1/(2n).

Original entry on oeis.org

4, 5, 6, 7, 8, 8, 9, 10, 10, 11, 11, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 22, 23, 23, 23, 23, 23, 24, 24, 24, 24, 25, 25, 25, 25, 25, 26, 26, 26, 26, 26, 27
Offset: 1

Views

Author

Clark Kimberling, Oct 05 2014

Keywords

Comments

For n > 1, let arch(n) = n*sin(Pi/n) be the Archimedean approximation to Pi (Finch, pp. 17 and 23) given by a regular polygon of n+1 sides. A248355 provides insight into the manner of convergence of arch(n) to Pi. (For the closely related function Arch, see A248347.)

Examples

			Approximations are shown here:
n    Pi - arch(n)      1/(2n)
1     3.14159...        0.5
2     1.14159...        0.25
3     0.543516...       0.16667
4     0.313166...       0.125
5     0.202666...       0.1
6     0.141593...       0.08333
7     0.105506...       0.07143
8     0.0801252...      0.0625
a(5) = 8 because Pi - arch(8) < 1/10 < Pi - arch(7).

Links

Crossrefs

Cf. A248356, A248357, A248358, A248347, A248578.

Programs

  • Mathematica
    z = 200; p[k_] := p[k] = k*Sin[Pi/k]; N[Table[Pi - p[n], {n, 1, z/10}]]
f[n_] := f[n] = Select[Range[z], Pi - p[#] < 1/(2 n) &, 1]
u = Flatten[Table[f[n], {n, 1, z}]]        (* A248355 *)
v = Flatten[Position[Differences[u], 0]]   (* A248356 *)
w = Flatten[Position[Differences[u], 1]]   (* A248357 *)
f = Table[Floor[1/(Pi - p[n])], {n, 1, z}] (* A248358 *)

Formula

a(n) ~ Pi*sqrt(Pi*n/3). - Vaclav Kotesovec, Oct 09 2014

A248357 Numbers k such that A248355(k+1) = A248355(k) + 1.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 11, 13, 16, 19, 21, 24, 28, 31, 34, 38, 42, 46, 51, 55, 60, 65, 70, 75, 81, 87, 93, 99, 105, 111, 118, 125, 132, 139, 147, 154, 162, 170, 178, 187, 195
Offset: 1

Views

Author

Clark Kimberling, Oct 05 2014

Keywords

Comments

The difference sequence of A248355 is (1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1,...), so that A248356 = (5, 8, 10, 12, 14, 15, ...) and A248357 = (1, 2, 3, 4, 6, 7, 9, 11, 13, 16, 19,...); A248356 and A248357 are a complementary pair.

Links

Crossrefs

Cf. A248355, A248356, A248358, A248347.

Programs

  • Mathematica
    z = 200; p[k_] := p[k] = k*Sin[Pi/k]; N[Table[Pi - p[n], {n, 1, z/10}]]
f[n_] := f[n] = Select[Range[z], Pi - p[#] < 1/(2 n) &, 1]
u = Flatten[Table[f[n], {n, 1, z}]]        (* A248355 *)
v = Flatten[Position[Differences[u], 0]]   (* A248356 *)
w = Flatten[Position[Differences[u], 1]]   (* A248357 *)
f = Table[Floor[1/(Pi - p[n])], {n, 1, z}] (* A248358 *)

A248356 Numbers k such that A248355(k+1) = A248355(k).

Original entry on oeis.org

5, 8, 10, 12, 14, 15, 17, 18, 20, 22, 23, 25, 26, 27, 29, 30, 32, 33, 35, 36, 37, 39, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 66, 67, 68, 69, 71, 72, 73, 74, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 88, 89, 90, 91, 92
Offset: 1

Views

Author

Clark Kimberling, Oct 05 2014

Keywords

Comments

The difference sequence of A248355 is (1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1,...), so that A248356 = (5, 8, 10, 12, 14, 15, ...) and A248357 = (1, 2, 3, 4, 6, 7, 9, 11, 13, 16, 19,...); A248356 and A248357 are a complementary pair.

Links

Crossrefs

Cf. A248355, A248357, A248358, A248347.

Programs

  • Mathematica
    z = 200; p[k_] := p[k] = k*Sin[Pi/k]; N[Table[Pi - p[n], {n, 1, z/10}]]
f[n_] := f[n] = Select[Range[z], Pi - p[#] < 1/(2 n) &, 1]
u = Flatten[Table[f[n], {n, 1, z}]]        (* A248355 *)
v = Flatten[Position[Differences[u], 0]]   (* A248356 *)
w = Flatten[Position[Differences[u], 1]]   (* A248357 *)
f = Table[Floor[1/(Pi - p[n])], {n, 1, z}] (* A248358 *)

A248578 a(n) = round(1/(Pi-n*sin(Pi/n))).

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 10, 12, 16, 19, 24, 28, 33, 38, 44, 50, 56, 63, 70, 77, 85, 94, 102, 112, 121, 131, 141, 152, 163, 174, 186, 198, 211, 224, 237, 251, 265, 280, 294, 310, 325, 341, 358, 375, 392, 410, 428, 446, 465, 484
Offset: 1

Views

Author

M. F. Hasler, Oct 08 2014

Keywords

Comments

n*sin(Pi/n) is known as the Archimedean approximation to Pi, the present sequence measures [the reciprocal of] the error. See A248358 for the integer part.

Crossrefs

Cf. A248358, A248355.

Programs

  • Mathematica
    Table[Round[1/(Pi -n Sin[Pi/n])],{n,50}] (* Harvey P. Dale, Jun 30 2022 *)
  • PARI
    a(n)=round(1/(Pi-n*sin(Pi/n)))
Showing 1-4 of 4 results.

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