A248355 -id:A248355 - cp's OEIS frontend

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

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 *)

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 *)

A248347 a(n) = floor(1/(Pi - 2^(n+1)*sin(Pi/2^(n+1)))).

Original entry on oeis.org

3, 12, 49, 198, 792, 3170, 12681, 50727, 202909, 811636, 3246545, 12986183, 51944732, 207778928, 831115713, 3324462855, 13297851421, 53191405684, 212765622737, 851062490950, 3404249963800, 13616999855201, 54467999420806, 217871997683226, 871487990732903

Offset: 1

Clark Kimberling, Oct 05 2014

Let Arch(n) = 2^(n+1)*sin(Pi/2^(n+1)) be the Archimedean approximation to Pi (Finch, pp. 17 and 23) given by a regular polygon of 2^(n+1) sides. A248347 provides insight into the manner of convergence of Arch(n) to Pi. Another provider is the fact that the least k for which Arch(k) < 1/4^n is A000027(n) = n. (For the closely related function arch, see A248355.)

			n    Pi - Arch(n)      1/(Pi - Arch(n))
1    0.313166...         3.1932...
2    0.0801252...       12.4805...
3    0.0201475...       49.6339...
4    0.00504416...     198.249...
5    0.0012615...      792.709...

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A000027, A248355, A248357, A248355, A248360.

z = 200; p[k_] := p[k] = (2^(k + 1))*Sin[Pi/2^(k + 1)]
Table[Floor[1/(Pi - p[n])], {n, 1, z}]  (* A248347  *)

a(n) ~ 6 * 4^(n+1) / Pi^3. - Vaclav Kotesovec, Oct 09 2014

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

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

M. F. Hasler, Oct 08 2014

nonn

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.

Cf. A248358, A248355.

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)))
```

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

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

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A248347 a(n) = floor(1/(Pi - 2^(n+1)*sin(Pi/2^(n+1)))).

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

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

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