A019952 -id:A019952 - cp's OEIS frontend

A381152 Decimal expansion of the isoperimetric quotient of a regular pentagon.

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

Offset: 0

Paolo Xausa, Feb 15 2025

The isoperimetric quotient of a closed curve is equal to 4*Pi*A/p^2, where A is the area enclosed by the curve and p is its perimeter. For a regular n-gon, this is equivalent to Pi/(n*tan(Pi/n)).

The isoperimetric quotient of a circle is 1.

			0.86480626597720996723118206585862333703828555690228...

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

Cf. A019952, A102771.

Cf. isoperimetric quotient of other regular polygons: A073010 (triangle), A003881 (square), A093766 (hexagon), A381153 (heptagon), A196522 (octagon), A381154 (9-gon), A381155 (10-gon), A381156 (11-gon), A381157 (12-gon).

First[RealDigits[Pi/(5*Tan[Pi/5]), 10, 100]]

Equals Pi/(5*tan(Pi/5)) = (Pi/5)*A019952.

Equals (4/25)*Pi*A102771.

A344172 Decimal expansion of 4sqrt(5 + 2sqrt(5)).

Original entry on oeis.org

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

Offset: 2

Wesley Ivan Hurt, May 10 2021

Decimal expansion of the volume of a rhombic triacontahedron with unit edge length.

			12.310734148701013610281...

Wikipedia, Rhombic triacontahedron

Cf. A344171 (rhombic triacontahedron surface area).
Cf. A344212 (rhombic triacontahedron midradius).
Cf. A019952 (rhombic triacontahedron radius of inscribed sphere).

RealDigits[4 Sqrt[5 + 2 Sqrt[5]], 10, 100][[1]] // Flatten

A153127 a(n) = (2n + 1)(5*n + 6).

Original entry on oeis.org

6, 33, 80, 147, 234, 341, 468, 615, 782, 969, 1176, 1403, 1650, 1917, 2204, 2511, 2838, 3185, 3552, 3939, 4346, 4773, 5220, 5687, 6174, 6681, 7208, 7755, 8322, 8909, 9516, 10143, 10790, 11457, 12144, 12851, 13578, 14325, 15092, 15879, 16686, 17513

Offset: 0

Reinhard Zumkeller, Dec 20 2008

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000566, A001622, A005408, A016861, A019952, A033571, A153126.

Table[(2n+1)(5n+6),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{6,33,80},50] (* Harvey P. Dale, Jun 07 2021 *)

a(n)=(2*n+1)*(5*n+6) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = A153126(2*n+1) = A000566(2*(n+1)) - 1.
a(n) = a(n-1) + 20*n + 7 (with a(0)=6). - Vincenzo Librandi, Dec 27 2010
G.f.: (-6-15*x+x^2)/(-1+x)^3 - Harvey P. Dale, Jun 07 2021
Sum_{n>=0} 1/a(n) = 5/7 - sqrt(1+2/sqrt(5))*Pi/14 - sqrt(5)*log(phi)/14 - 5*log(5)/28 + 2*log(2)/7, where phi is the golden ratio (A001622). - Amiram Eldar, Aug 23 2022
From Elmo R. Oliveira, Oct 27 2024: (Start)
E.g.f.: exp(x)*(6 + 27*x + 10*x^2).
a(n) = A005408(n)*A016861(n+1).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A190987 a(n) = 10a(n-1) - 5a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 10, 95, 900, 8525, 80750, 764875, 7245000, 68625625, 650031250, 6157184375, 58321687500, 552430953125, 5232701093750, 49564856171875, 469485056250000, 4447026281640625, 42122837535156250, 398993243943359375, 3779318251757812500, 35798216297861328125

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-5).

Cf. A190958 (index to generalized Fibonacci sequences).

Cf. A019934 (sqrt(5-2*sqrt(5))), A019952 (sqrt(5+2*sqrt(5))).

[Round(5^((n-1)/2)*Evaluate(ChebyshevU(n), Sqrt(5))): n in [0..30]]; // G. C. Greubel, Sep 07 2022

Mathematica
```
LinearRecurrence[{10,-5}, {0,1}, 50]
```

A190987 = BinaryRecurrenceSequence(10, -5, 0, 1)
[A190987(n) for n in (0..30)] # G. C. Greubel, Sep 07 2022

G.f.: x/(1 - 10*x + 5*x^2). - Philippe Deléham, Oct 12 2011

E.g.f.: (1/(2*sqrt(5)))*exp(5*x)*sinh(2*sqrt(5)*x). - G. C. Greubel, Sep 07 2022

A131895 a(n) = (n + 2)(5n + 1)/2.

Original entry on oeis.org

1, 9, 22, 40, 63, 91, 124, 162, 205, 253, 306, 364, 427, 495, 568, 646, 729, 817, 910, 1008, 1111, 1219, 1332, 1450, 1573, 1701, 1834, 1972, 2115, 2263, 2416, 2574, 2737, 2905, 3078, 3256, 3439, 3627, 3820, 4018, 4221, 4429, 4642, 4860, 5083, 5311, 5544

Offset: 0

Gary W. Adamson, Jul 24 2007

Row sums of triangle A131894.

Binomial transform of (1, 8, 5, 0, 0, 0, ...).

			a(2) = 22 = sum of row 2 terms of triangle A131894: (11 + 6 + 5).
a(2) = 22 = (1, 2, 1) dot (1, 8, 5) = (1 + 16 + 5).

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001622, A019952, A131894.

A131895:=n->(n+2)*(5*n+1)/2; seq(A131895(n), n=0..50); # Wesley Ivan Hurt, Mar 26 2014

LinearRecurrence[{3,-3,1},{1,9,22},50] (* Harvey P. Dale, Sep 11 2015 *)

a(n)=(n+2)*(5*n+1)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = a(n-1) + 5*n + 3 (with a(0)=1). - Vincenzo Librandi, Nov 23 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=9, a(2)=22. - Harvey P. Dale, Sep 11 2015
From Elmo R. Oliveira, Oct 22 2024: (Start)
G.f.: (1 + 6*x - 2*x^2)/(1 - x)^3.
E.g.f.: (1 + 8*x + 5*x^2/2)*exp(x). (End)
Sum_{n>=0} 1/a(n) = (2 + sqrt(1+2/sqrt(5))*Pi + sqrt(5)*log(phi) + 5*log(5)/2)/9, where phi is the golden ratio (A001622). - Amiram Eldar, Jun 02 2025

More terms from Vladimir Joseph Stephan Orlovsky, Dec 04 2008

Simpler definition from Wesley Ivan Hurt, Mar 26 2014

A348757 Decimal expansion of the area of a regular pentagram inscribed in a unit-radius circle.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Nov 12 2021

An algebraic number of degree 4. The smaller of the two positive roots of the equation 16*x^4 - 2500*x^2 + 3125 = 0.

			1.12256994144896343110486287949381696894803120580270...

Robert B. Banks, Slicing Pizzas, Racing Turtles, and Further Adventures in Applied Mathematics, Princeton University Press, 2012, p. 15.

Herta T. Freitag, Problem 3855, School Science and Mathematics, Vol. 81, No. 4 (1981), p. 352; Solution to Problem 3855 by David A. Blaeuer, ibid., Vol. 82, No. 3 (1982), pp. 265-266.
Mathematics Stack Exchange, Area of a five pointed star, 2014.
Wikipedia, Pentagram.
Index entries for algebraic numbers, degree 4

Cf. A001622, A019845, A019916, A019952, A019970, A094874, A104955, A134974, A179050, A188593, A344172.

RealDigits[5*Sin[Pi/5]/GoldenRatio^2, 10, 100][[1]]

Equals 5*sin(Pi/5)/phi^2, where phi is the golden ratio (A001622).
Equals 5/(cot(Pi/5) + cot(Pi/10)).
Equals 10*tan(Pi/10)/(3 - tan(Pi/10)^2).
Equals (5/2)*sqrt((25 -11*sqrt(5))/2).
Equals 5*(5 - sqrt(5))/(4*sqrt(5 + 2*sqrt(5))) = A094874 * A179050 = 10 * A094874 / A344172.

A357715 Decimal expansion of sqrt(16 + 32 / sqrt(5)).

Original entry on oeis.org

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

Offset: 1

Michal Paulovic, Oct 10 2022

The perimeter of a golden rectangle inscribed in a unit circle.
The width and height of the rectangle are:
  W = sqrt(2 - 2 / sqrt(5)) = A179290.
  H = sqrt(2 + 2 / sqrt(5)) = A121570.

			5.5055276818846941...

Cf. A019934, A019952, A019970, A121570, A179290, A204188, A356869.

Maple
```
sqrt(16 + 32 / sqrt(5));
```
Mathematica
```
Sqrt[16 + 32/Sqrt[5]]
```
PARI
```
sqrt(16 + 32 / sqrt(5))
```

Equals (4 / sqrt(5)) * sqrt(5 + 2 * sqrt(5)) = A356869 * A019970.
Equals sqrt(5 + 2 * sqrt(5)) / (sqrt(5) / 4) = A019970 / A204188.
Equals 4 * sqrt(1 + 2 / sqrt(5)) = 4 * A019952.
Equals 4 / sqrt(5 - 2 * sqrt(5)) = 4 / A019934.

cp's OEIS Frontend

A381152 Decimal expansion of the isoperimetric quotient of a regular pentagon.

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A344172 Decimal expansion of 4*sqrt(5 + 2*sqrt(5)).

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A153127 a(n) = (2*n + 1)*(5*n + 6).

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A190987 a(n) = 10*a(n-1) - 5*a(n-2), with a(0)=0, a(1)=1.

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A131895 a(n) = (n + 2)*(5*n + 1)/2.

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A348757 Decimal expansion of the area of a regular pentagram inscribed in a unit-radius circle.

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A357715 Decimal expansion of sqrt(16 + 32 / sqrt(5)).

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Maple

Mathematica

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A344172 Decimal expansion of 4sqrt(5 + 2sqrt(5)).

A153127 a(n) = (2n + 1)(5*n + 6).

A190987 a(n) = 10a(n-1) - 5a(n-2), with a(0)=0, a(1)=1.

A131895 a(n) = (n + 2)(5n + 1)/2.