A248347 -id:A248347 - cp's OEIS frontend

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

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

Clark Kimberling, Oct 05 2014

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.)

			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).

Clark Kimberling, Table of n, a(n) for n = 1..10000

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

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 *)

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

Clark Kimberling, Oct 05 2014

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.

Clark Kimberling, Table of n, a(n) for n = 1..300

Cf. A248355, A248356, A248358, A248347.

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 *)

A248358 Floor(1/(Pi - n*sin(Pi/n))).

Original entry on oeis.org

0, 0, 1, 3, 4, 7, 9, 12, 15, 19, 23, 27, 32, 38, 43, 49, 56, 62, 69, 77, 85, 93, 102, 111, 121, 130, 141, 151, 162, 174, 186, 198, 210, 223, 237, 250, 265, 279, 294, 309, 325, 341, 357, 374, 391, 409, 427, 445, 464, 483, 503, 523, 543, 564, 585, 606, 628

Offset: 1

Clark Kimberling, Oct 05 2014

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. A248358 and A248355 provide insight into the manner of convergence of arch(n) to Pi. (For the closely related function Arch, see A248347.)

See A248578 for the similar sequence round(1/(Pi-n*sin(Pi/n))). - M. F. Hasler, Oct 08 2014

			n    Pi - arch(n)    1/(Pi - arch(n))
1     3.14159...       0.3183...
2     1.14159...       0.8759...
3     0.54351...       1.8398...
4     0.31316...       3.1932...
5     0.20266...       4.9342...
6     0.14159...       7.0625...

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A248355, A248356, A248357, A248347, A248578.

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 *)

a(n)=1\(Pi-n*sin(Pi/n)) \\ M. F. Hasler, Oct 08 2014

a(n) ~ 6*n^2/Pi^3. - Vaclav Kotesovec, Oct 09 2014

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

Clark Kimberling, Oct 05 2014

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.

Clark Kimberling, Table of n, a(n) for n = 1..9000

Cf. A248355, A248357, A248358, A248347.

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 *)

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

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

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A248358 Floor(1/(Pi - n*sin(Pi/n))).

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

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