A019916 -id:A019916 - cp's OEIS frontend

A022521 a(n) = (n+1)^5 - n^5.

1, 31, 211, 781, 2101, 4651, 9031, 15961, 26281, 40951, 61051, 87781, 122461, 166531, 221551, 289201, 371281, 469711, 586531, 723901, 884101, 1069531, 1282711, 1526281, 1803001, 2115751, 2467531

Offset: 0

N. J. A. Sloane

Last digit of a(n) is always 1. Last two digits of a(n) (i.e., a(n) mod 100) are repeated periodically with palindromic part of period 20 {1,31,11,81,1,51,31,61,81,51,51,81,61,31,51,1,81,11,31,1}. Last three digits of a(n) (i.e., a(n) mod 1000) are repeated periodically with palindromic part of period 200. - Alexander Adamchuk, Aug 11 2006
In Conway and Guy, these numbers are called nexus numbers of order 5. - M. F. Hasler, Jan 27 2013
Numbers that can be arranged in a triangular-antitegmatic icosachoron (the 4D version of "rhombic dodecahedal numbers" (A005917)). - Steven Lu, Mar 28 2023

John H. Conway and Richard K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 54.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Polytope Wiki, Triangular-antitegmatic Icosachoron
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

First differences of A000584.
Column k=4 of array A047969.
Cf. A003215, A005917, A006322, A008292, A019916, A019952, A022522.

[(n+1)^5-n^5: n in [0..30]]; // Vincenzo Librandi, Nov 23 2011

Table[(n+1)^5-n^5, {n,0,30}] (* Vincenzo Librandi, Nov 23 2011 *)

a(n)=(n+1)^5-n^5 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = A003215(n) + 24 * A006322(n). - Xavier Acloque, Oct 11 2003
G.f.: (-1-x^4-26*x^3-66*x^2-26*x)/(x-1)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009
G.f.: polylog(-5, x)*(1-x)/x. See the g.f. of the rows of A008292 by Vladeta Jovovic, Sep 02 2002. - Wolfdieter Lang, May 10 2021
Sum_{n>=0} 1/a(n) = c1*tanh(c2/2) - c2*tanh(c1/2), where c1 = tan(3*Pi/10)*Pi and c2 = tan(Pi/10)*Pi. - Amiram Eldar, Jan 27 2022

A375068 Decimal expansion of the sagitta of a regular pentagon with unit side length.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Jul 29 2024

			0.1624598481164531630779357061075672324774517357607...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Regular Polygon.
Eric Weisstein's World of Mathematics, Sagitta
Index entries for algebraic numbers, degree 4.

Cf. A300074 (circumradius), A375067 (apothem), A102771 (area).
Cf. sagitta of other polygons with unit side length: A020769 (triangle), A174968 (square), A375069 (hexagon), A374972 (heptagon), A375070 (octagon), A375153 (9-gon), A375189 (10-gon), A375192 (11-gon), A375194 (12-gon).
Cf. A019916, A179050, A229760.

First[RealDigits[Tan[Pi/10]/2, 10, 100]]

tan(Pi/10)/2 \\ Charles R Greathouse IV, Feb 04 2025

polrootsreal(80*x^4-40*x^2+1)[3] \\ Charles R Greathouse IV, Feb 04 2025

Equals tan(Pi/10)/2 = sqrt(1-2/sqrt(5))/2 = A019916/2.
Equals A300074 - A375067.
Equals A179050/5 = sqrt(A229760)/10. - Hugo Pfoertner, Jul 30 2024

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.

A343057 Decimal expansion of tan(Pi/32).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Apr 04 2021

			0.098491403357164253077197...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Wikipedia, Trigonometric constants expressed in real radicals
Index entries for algebraic numbers, degree 8.

Cf. A343055 (sin(Pi/32)), A343056 (cos(Pi/32)).

tan(Pi/m): A002194 (m=3), A019934 (m=5), A020760 (m=6), A343058 (m=7), A188582 (m=8), A019918 (m=9), A019916 (m=10), A019913 (m=12), A343059 (m=14), A019910 (m=15), A343060 (m=16), A343061 (m=17), A019908 (m=18), A019907 (m=20), A343062 (m=24), A019904 (m=30), A343057 (m=32), A019903 (m=36).

R:= RealField(125); Tan(Pi(R)/32); // G. C. Greubel, Sep 30 2022

RealDigits[Tan[Pi/32], 10, 120, -1][[1]] (* Amiram Eldar, Apr 27 2021 *)

PARI
```
tan(Pi/32)
```

sqrt((2-sqrt(2+sqrt(2+sqrt(2))))/(2+sqrt(2+sqrt(2+sqrt(2)))))

numerical_approx(tan(pi/32), digits=125) # G. C. Greubel, Sep 30 2022

Equals sqrt( (2-sqrt(2+sqrt(2+sqrt(2))))/(2+sqrt(2+sqrt(2+sqrt(2)))) ).

A341859 Decimal expansion of 4 - (8/5)*sqrt(5).

Original entry on oeis.org

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

Offset: 0

Gleb Koloskov, Mar 07 2021

In a triangle inscribed in a unit circle this is the maximal value of its inradius, such that a minimal closed Steiner chain of circles (10 circles) can be sandwiched between the incircle and circumcircle of the triangle.
It can be found as follows.
The squared distance between the centers of the two chain-defining circles is known to be d^2 = (R-r)^2 - 4*r*R*tan(Pi/n)^2.
On the other hand, the squared distance between the circumcenter and the incenter of triangle is known to be d^2 = R*(R-2*r).
Thus, in order to make a valid closed chain of circles, the inradius of triangle inscribed in the unit circle must be equal to 4*tan(Pi/n)^2.
Given that the maximum of such inradius is 0.5, the minimal number of chained circles is n=10, which gives the maximal value r = 4*tan(Pi/10)^2 = 0.42... < 0.5.

			0.4222912360003364857453221300299580232950106246215588411665644073...

Liang-Shin Hahn. Complex Numbers and Geometry (Mathematical Association of America Textbooks). The Mathematical Association of America, 1994, 140-141.

Wikipedia, Steiner Chain.

Cf. A019916, A001622.

RealDigits[4*Tan[18 Degree]^2, 10, 120][[1]]

PARI
```
4-8/5*sqrt(5)
```

Equals 4*A019916^2 = 4*tan(Pi/10)^2 = 4 - (8/5)*sqrt(5) = (4/5)*(7 - 4*phi) = (4/5)*(7 - 4*A001622), where phi is the golden ratio from A001622.

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.

cp's OEIS Frontend

A022521 a(n) = (n+1)^5 - n^5.

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A375068 Decimal expansion of the sagitta of a regular pentagon with unit side length.

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

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A343057 Decimal expansion of tan(Pi/32).

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A341859 Decimal expansion of 4 - (8/5)*sqrt(5).

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

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