A003881 -id:A003881 - cp's OEIS frontend

A381153 Decimal expansion of the isoperimetric quotient of a regular heptagon.

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

Offset: 0

Paolo Xausa, Feb 15 2025

For the definition of isoperimetric quotient, see A381152.

			0.93194062349909574595222630089422754574528525154715...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Isoperimetric Quotient.
Wikipedia, Isoperimetric inequality.

Cf. A178817, A343058.

Cf. isoperimetric quotient of other regular polygons: A073010 (triangle), A003881 (square), A381152 (pentagon), A093766 (hexagon), A196522 (octagon), A381154 (9-gon), A381155 (10-gon), A381156 (11-gon), A381157 (12-gon).

First[RealDigits[Pi/(7*Tan[Pi/7]), 10, 100]]

Equals Pi/(7*tan(Pi/7)) = Pi/(7*A343058).

Equals (4/49)*Pi*A178817.

A381154 Decimal expansion of the isoperimetric quotient of a regular 9-gon.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Feb 15 2025

For the definition of isoperimetric quotient, see A381152.

			0.959050541873609358074543306708643413020181580975...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Isoperimetric Quotient.
Wikipedia, Isoperimetric inequality.

Cf. A019918, A256853.

Cf. isoperimetric quotient of other regular polygons: A073010 (triangle), A003881 (square), A381152 (pentagon), A093766 (hexagon), A381153 (heptagon), A196522 (octagon), A381155 (10-gon), A381156 (11-gon), A381157 (12-gon).

First[RealDigits[Pi/(9*Tan[Pi/9]), 10, 100]]

Equals Pi/(9*tan(Pi/9)) = Pi/(9*A019918).

Equals (4/81)*Pi*A256853.

A381155 Decimal expansion of the isoperimetric quotient of a regular 10-gon.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Feb 15 2025

For the definition of isoperimetric quotient, see A381152.

			0.96688279904640254032818321918275298846986824108440...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Isoperimetric Quotient.
Wikipedia, Isoperimetric inequality.

Cf. A019916, A178816.

Cf. isoperimetric quotient of other regular polygons: A073010 (triangle), A003881 (square), A381152 (pentagon), A093766 (hexagon), A381153 (heptagon), A196522 (octagon), A381154 (9-gon), A381156 (11-gon), A381157 (12-gon).

First[RealDigits[Pi/(10*Tan[Pi/10]), 10, 100]]

Equals Pi/(10*tan(Pi/10)) = Pi/(10*A019916).

Equals (1/25)*Pi*A178816.

A381156 Decimal expansion of the isoperimetric quotient of a regular 11-gon.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Feb 15 2025

For the definition of isoperimetric quotient, see A381152.

			0.97266200091990681953828897938526763171296541114234...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Isoperimetric Quotient.
Wikipedia, Isoperimetric inequality.

Cf. A256854.

Cf. isoperimetric quotient of other regular polygons: A073010 (triangle), A003881 (square), A381152 (pentagon), A093766 (hexagon), A381153 (heptagon), A196522 (octagon), A381154 (9-gon), A381155 (10-gon), A381157 (12-gon).

First[RealDigits[Pi/(11*Tan[Pi/11]), 10, 100]]

Equals Pi/(11*tan(Pi/11)).

Equals (4/121)*Pi*A256854.

A381157 Decimal expansion of the isoperimetric quotient of a regular 12-gon.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Feb 15 2025

For the definition of isoperimetric quotient, see A381152.

			0.97704861665685333572562679495712274710387812858570...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Isoperimetric Quotient.
Wikipedia, Isoperimetric inequality.

Cf. A019913, A178809.

Cf. isoperimetric quotient of other regular polygons: A073010 (triangle), A003881 (square), A381152 (pentagon), A093766 (hexagon), A381153 (heptagon), A196522 (octagon), A381154 (9-gon), A381155 (10-gon), A381156 (11-gon).

First[RealDigits[Pi/(12*Tan[Pi/12]), 10, 100]]

Equals Pi/(12*tan(Pi/12)) = Pi/(12*A019913).

Equals (1/36)*Pi*A178809.

A096955 Denominators of rational approximation to Pi/4 from Machins's formula.

Original entry on oeis.org

1195, 1706489875, 12184551018734375, 24359780855939418203125, 104359128170408663038552734375, 1639301884061026141391921953564453125, 30432532948821209122295591520605416259765625

Offset: 0

Wolfdieter Lang, Jul 23 2004

Machin's formula: Pi/4 = 4*arctan(1/5) - arctan(1/239).

Numerators are given in A096954.

			A096954(7)/a(7) =
170660807873601670198453967268421248219727522686104 /217292089321202035784330810406062747771759033203125
= 0.78539816339715...

W. Walter, Analysis I (in German), Springer, 3. Auflage, 1992; p. 216.

Wolfdieter Lang, More comments.
Eric Weisstein's World of Mathematics, Machin's Formula.

Cf. A003881, A096954.

a(n) = denominator(M(n)), with M(n)=4*arctan(1/5, n) - arctan(1/239, n) with arctan(x, n):=sum((((-1)^k)*x^(2k+1))/(2*k+1), k=0..n).

A136485 Number of unit square lattice cells enclosed by origin centered circle of diameter n.

Original entry on oeis.org

0, 0, 4, 4, 12, 16, 24, 32, 52, 60, 76, 88, 112, 120, 148, 164, 192, 216, 256, 276, 308, 332, 376, 392, 440, 476, 524, 556, 608, 648, 688, 732, 796, 832, 904, 936, 1012, 1052, 1124, 1176, 1232, 1288, 1372, 1428, 1508, 1560, 1648, 1696, 1788, 1860, 1952, 2016

Offset: 1

Glenn C. Foster (gfoster(AT)uiuc.edu), Jan 02 2008

a(n) is the number of complete squares that fit inside the circle with diameter n, drawn on squared paper.

			a(3) = 4 because a circle centered at the origin and of radius 3/2 encloses (-1,-1), (-1,1), (1,-1), (1,1).

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A003881, A136483, A136513.

Alternating merge of A119677 of A136485.

A136485:= func< n | n le 1 select 0 else 4*(&+[Floor(Sqrt((n/2)^2-j^2)): j in [1..Floor(n/2)]]) >;
[A136485(n): n in [1..100]]; // G. C. Greubel, Jul 29 2023

Table[4*Sum[Floor[Sqrt[(n/2)^2 - k^2]], {k,Floor[n/2]}], {n,100}]

def A136485(n): return 4*sum(floor(sqrt((n/2)^2-k^2)) for k in range(1,(n//2)+1))
[A136485(n) for n in range(1,101)] # G. C. Greubel, Jul 29 2023

a(n) = 4 * Sum_{k=1..floor(n/2)} floor(sqrt((n/2)^2 - k^2)).
a(n) = 4 * A136483(n).
a(n) = 2 * A136513(n).
Lim_{n -> oo} a(n)/(n^2) -> Pi/4 (A003881).
a(n) = [x^(n^2)] (theta_3(x^4) - 1)^2 / (1 - x). - Ilya Gutkovskiy, Nov 24 2021

A152584 Decimal expansion of (Pi^3)/24.

Original entry on oeis.org

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

Offset: 1

Eric Desbiaux, Dec 08 2008

Consider infinite sum made of areas of circles Pi*radius^2 with diameter 1/n.
The volume is (Pi/4)*(1 + 1/4 + 1/9 + 1/16 + 1/25 + ... + 1/n^2)
= (1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ...)*(1 + 1/4 + 1/9 + 1/16 + 1/25 + ...)
= (Pi/4) * (Pi^2/6) = Pi^3/24.
Equals volume of a cone of height Pi^2/8 and radius 1.
Equals volume of a sphere (4*Pi*Pi^2/32)/3 with radius^3 = (Pi^2/32).

			1.291928195012492507311513127795891466759387023578...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for transcendental numbers

Cf. A013661, A003881, A102753, A019679, A111003.

RealDigits[(Pi^3)/24, 10, 50][[1]] (* G. C. Greubel, Jun 09 2017 *)

PARI
```
(Pi^3)/24 \\ G. C. Greubel, Jun 09 2017
```

Equals Integral_{x=0..oo} arctan(x)^2/(x^2 + 1) dx. - Amiram Eldar, Aug 06 2020

A197758 Decimal expansion of least x>0 having sin(2x)=4*sin(8x).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Oct 19 2011

For a discussion and guide to related sequences, see A197739.

			x=0.37145829403348635250583272851951240980...

Cf. A197739

b = 1; c = 4;
f[x_] := Cos[b*x]^2; g[x_] := Sin[c*x]^2; s[x_] := f[x] + g[x];
r = x /. FindRoot[b*Sin[2 b*x] == c*Sin[2 c*x], {x, .37, .38}, WorkingPrecision -> 110]
RealDigits[r]  (* A197758 *)
m = s[r]
RealDigits[m]  (* A197759 *)
Plot[{b*Sin[2 b*x], c*Sin[2 c*x]}, {x, 0, Pi}]
d = m/2; t = x /. FindRoot[s[x] == d, {x, 0.64, 0.65}, WorkingPrecision -> 110]
RealDigits[t]  (* A197760 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
d = m/3; t = x /. FindRoot[s[x] == d, {x, 0.72, 0.73}, WorkingPrecision -> 110]
RealDigits[t]  (* A197761 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
d = 1; t = x /. FindRoot[s[x] == d, {x, 0.6, 0.7}, WorkingPrecision -> 110]
RealDigits[t]  (* A019692, pi/5 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
d = 1/2; t = x /. FindRoot[s[x] == d, {x, 0.6, 0.8}, WorkingPrecision -> 110]
RealDigits[t]   (* A003881 *)
Plot[{s[x], d}, {x, 0, 1}, AxesOrigin -> {0, 0}]
RealDigits[ ArcTan[ Sqrt[ Root[17#^3 - 109#^2 + 115# - 15&, 1] ] ], 10, 99] // First (* Jean-François Alcover, Feb 27 2013 *)

A281964 Real part of n!*Sum_{k=1..n} i^(k-1)/k, where i is sqrt(-1).

Original entry on oeis.org

1, 2, 4, 16, 104, 624, 3648, 29184, 302976, 3029760, 29698560, 356382720, 5111976960, 71567677440, 986336870400, 15781389926400, 289206418636800, 5205715535462400, 92506221468057600, 1850124429361152000, 41285515024760832000, 908281330544738304000

Offset: 1

Daniel Suteu, Feb 06 2017

nonn

			For n=5, a(5) = 104, which is the real part of 5!*(1/1 + i/2 - 1/3 - i/4 + 1/5) = 104+30*i.

Iain Fox, Table of n, a(n) for n = 1..450 (first 100 terms from Daniel Suteu)

The corresponding imaginary part is A282132.

Cf. A003881, A024167.

a(n) = real(n!*sum(k=1, n, I^(k-1)/k));

first(n) = x='x+O('x^(n+1)); Vec(serlaplace(atan(x)/(1 - x))) \\ Iain Fox, Dec 19 2017

a(n) ~ Pi/4 * n!.
a(1) = 1, a(n+1) = a(n)*(n+1) + n!*cos(Pi*n/2).
E.g.f.: arctan(x)/(1 - x). - Ilya Gutkovskiy, Dec 19 2017

cp's OEIS Frontend

A381153 Decimal expansion of the isoperimetric quotient of a regular heptagon.

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A381154 Decimal expansion of the isoperimetric quotient of a regular 9-gon.

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A381155 Decimal expansion of the isoperimetric quotient of a regular 10-gon.

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A381156 Decimal expansion of the isoperimetric quotient of a regular 11-gon.

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A381157 Decimal expansion of the isoperimetric quotient of a regular 12-gon.

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A096955 Denominators of rational approximation to Pi/4 from Machins's formula.

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A136485 Number of unit square lattice cells enclosed by origin centered circle of diameter n.

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A152584 Decimal expansion of (Pi^3)/24.