A019952 -id:A019952 - 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

A033571 a(n) = (2n + 1)(5*n + 1).

Original entry on oeis.org

1, 18, 55, 112, 189, 286, 403, 540, 697, 874, 1071, 1288, 1525, 1782, 2059, 2356, 2673, 3010, 3367, 3744, 4141, 4558, 4995, 5452, 5929, 6426, 6943, 7480, 8037, 8614, 9211, 9828, 10465, 11122, 11799, 12496, 13213, 13950, 14707, 15484, 16281, 17098, 17935, 18792, 19669, 20566, 21483

Offset: 0

N. J. A. Sloane

Sequence found by reading the line from 1, in the direction 1, 18, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. This is one of the diagonals in the spiral. - Omar E. Pol, Sep 10 2011

Also sequence found by reading the line from 1, in the direction 1, 18, ..., in the square spiral whose edges have length A195013 and whose vertices are the numbers A195014. This is a line perpendicular to the main axis A195015 in the same spiral. - Omar E. Pol, Oct 14 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000
Leo Tavares, Illustration: Stellar Layers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A153127. - Reinhard Zumkeller, Dec 20 2008
Cf. A000566, A008596, A010010, A153126, A158186.
Cf. A001622, A003154, A007742, A019952.

List([0..50], n-> (2*n+1)*(5*n+1)); # G. C. Greubel, Oct 12 2019

[(2*n+1)*(5*n+1): n in [0..50]]; // G. C. Greubel, Oct 12 2019

seq((2*n+1)*(5*n+1), n=0..50); # G. C. Greubel, Oct 12 2019

Table[(2*n+1)*(5*n+1), {n,0,50}] (* G. C. Greubel, Oct 12 2019 *)

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

[(2*n+1)*(5*n+1) for n in range(50)] # G. C. Greubel, Oct 12 2019

a(n) = A153126(2*n) = A000566(2*n+1). - Reinhard Zumkeller, Dec 20 2008
From Reinhard Zumkeller, Mar 13 2009: (Start)
a(n) = A008596(n) + A158186(n), for n > 0.
a(n) = A010010(n) - A158186(n). (End)
a(n) = a(n-1) + 20*n - 3 (with a(0)=1). - Vincenzo Librandi, Nov 17 2010
From G. C. Greubel, Oct 12 2019: (Start)
G.f.: (1 + 15*x + 4*x^2)/(1-x)^3.
E.g.f.: (1 + 17*x + 10*x^2)*exp(x). (End)
a(n) = A003154(n+1) + A007742(n). - Leo Tavares, Mar 27 2022
Sum_{n>=0} 1/a(n) = sqrt(1+2/sqrt(5))*Pi/6 + sqrt(5)*log(phi)/6 + 5*log(5)/12 - 2*log(2)/3, where phi is the golden ratio (A001622). - Amiram Eldar, Aug 23 2022

Terms a(36) onward added by G. C. Greubel, Oct 12 2019

A344212 Decimal expansion of 1 + 1/sqrt(5).

Original entry on oeis.org

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

Offset: 1

Wesley Ivan Hurt, May 11 2021

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

Essentially the same sequence of digits as A176453, A134974, A020762 and A010476. - R. J. Mathar, May 16 2021

			1.447213595499957939281834733746255247088123671922305...

Wikipedia, Rhombic triacontahedron.
Index entries for algebraic numbers, degree 2.

Cf. A019952 (rhombic triacontahedron inscribed sphere radius).
Cf. A344171 (rhombic triacontahedron surface area).
Cf. A344172 (rhombic triacontahedron volume).
Cf. A010476, A020762, A081005, A134974, A176453, A187798, A242671.

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

5^-.5+1 \\ Charles R Greathouse IV, Feb 04 2025

polrootsreal(5*x^2-10*x+4)[2] \\ Charles R Greathouse IV, Feb 04 2025

From Amiram Eldar, Nov 28 2024: (Start)
Equals 2*A242671 = 1/A187798.
Equals Product_{k>=0} (1 + 1/A081005(k)). (End)

A019934 Decimal expansion of tangent of 36 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Also the decimal expansion of cotangent of 54 degrees. - Mohammad K. Azarian, Jun 30 2013

A quartic integer. - Charles R Greathouse IV, Aug 27 2017

			0.72654252800536088589546675748061874961609239296520...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Exact trigonometric constants.
Index entries for algebraic numbers, degree 4.

Cf. A019934, A019952, A019970, A002163, A063226.

RealDigits[Tan[36 Degree],10,120][[1]] (* Harvey P. Dale, Nov 06 2012 *)

tan(Pi/5) \\ Charles R Greathouse IV, Aug 27 2017

This number is sqrt(5-2*sqrt(5)). This number * A019970 = sqrt(5) = A002163. - R. J. Mathar, Jun 18 2006
The smallest positive solution of cos(4*arctan(x)) = cos(6*arctan(x)). - Thomas Olson, Oct 03 2014
Let r(n) = (n - 1)/(n + 1) if n mod 4 = 1, (n + 1)/(n - 1) otherwise; then this constant (A019934) equals with Product_{n>=0} r(10*n+5) = (2/3) * (8/7) * (12/13) * (18/17) * ... - Dimitris Valianatos, Sep 14 2019
Equals Product_{k>=1} (1 + (-1)^k/A063226(k)). - Amiram Eldar, Nov 23 2024
Equals 1/A019952. - Hugo Pfoertner, Nov 23 2024
tan(Pi/5) = A019845 / A019863. - R. J. Mathar, Aug 31 2025
Smallest positive of the 4 real-valued roots of x^4-10*x^2+5=0. (Other A019970). - R. J. Mathar, Aug 31 2025

A200135 Decimal expansion of the negated value of the digamma function at 1/5.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Nov 13 2011

			Psi(1/5) =  -5.289039896592188295547207962...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Wikipedia, Digamma function
Index entries for sequences related to the digamma function

Cf. A020759, A047787, A020777, A200064, A200134, A200136, A200137, A200138.

SetDefaultRealField(RealField(100)); R:= RealField(); -EulerGamma(R) -Pi(R)*Sqrt(1+2/Sqrt(5))/2 -5*Log(5)/4 -Sqrt(5)/4*Log((3+Sqrt(5)/2) ); // G. C. Greubel, Sep 03 2018

-gamma-Pi*sqrt(1+2/sqrt(5))/2-5*log(5)/4-sqrt(5)/4*log((3+sqrt(5)/2) ); evalf(%) ;

RealDigits[-PolyGamma[1/5], 10, 105] // First (* Jean-François Alcover, Feb 11 2013 *)

-psi(1/5) \\ Charles R Greathouse IV, Jul 19 2013

Psi(1/5) = -gamma - Pi*sqrt(1 + 2/sqrt(5))/2 - 5*log(5)/4 -sqrt(5)*log((3 + sqrt(5))/2)/4 where gamma = A001620, sqrt(1 + 2/sqrt(5)) = A019952, (3 + sqrt(5))/2 = A104457.

More terms from Jean-François Alcover, Feb 11 2013

A237603 Decimal expansion of the inscribed sphere radius in a regular dodecahedron with unit edge.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Feb 25 2014

Equals phi^2/(2*xi), where phi is the golden ratio (A001622, 2*cos(Pi/5)) and xi is its associate (A182007, 2*sin(Pi/5)).

			1.1135163644116067351943750394869493758831503698864877726012080...

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 451.

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 4.

Cf. A001622, A182007, A019863, A019863, A019952, A374771 (sphere volume).

Cf. Platonic solids inradii: A020781 (tetrahedron), A020763 (octahedron), A179294 (icosahedron).

RealDigits[ Cos[Pi/5]^2 / Sin[Pi/5], 10, 111][[1]] (* Or *)
RealDigits[ Sqrt[5/8 + 11/(8 Sqrt[5])], 10, 111][[1]] (* Robert G. Wilson v, Feb 28 2014 *)

PARI
```
sqrt(250+110*sqrt(5))/20
```

Equals A001622^2/A182007 = (cos(Pi/5))^2/sin(Pi/5) = A019863^2/A019845 = cos(Pi/5)*cotan(Pi/5) = A019863*A019952 = 1/sin(Pi/5) - sin(Pi/5) = A019845^(-1) - A019845 = sqrt(250+110*sqrt(5))/20.

A375067 Decimal expansion of the apothem (inradius) of a regular pentagon with unit side length.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Jul 29 2024

			0.688190960235586769103604790955443839762949668...

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

Cf. A300074 (circumradius), A375068 (sagitta), A102771 (area).
Cf. apothem of other polygons with unit side length: A020769 (triangle), A020761 (square), A010527 (hexagon), A374971 (heptagon), A174968 (octagon), A375152 (9-gon), A179452 (10-gon), A375191 (11-gon), A375193 (12-gon).
Cf. A019934, A019952, A019863, A131595.

Mathematica
```
First[RealDigits[Cot[Pi/5]/2, 10, 100]]
```

 5/tan(Pi/5) \\ Charles R Greathouse IV, Feb 05 2025

Equals cot(Pi/5)/2 = A019952/2.
Equals 1/(2*tan(Pi/5)) = 1/(2*A019934).
Equals sqrt(1/4 + 1/(2*sqrt(5))).
Equals (1/2)*csc(Pi/5)*cos(Pi/5) = A300074*A019863.
Equals A300074 - A375068.
Equals A131595/30. - Hugo Pfoertner, Jul 30 2024

A019916 Decimal expansion of tan(Pi/10) (angle of 18 degrees).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

In a regular pentagon inscribed in a unit circle this is the cube of the length of the side divided by 5: (1/5)*(sqrt(3 - phi))^3 with phi from A001622. - Wolfdieter Lang, Jan 08 2018
Quartic number of denominator 5 and minimal polynomial 5x^4 - 10x^2 + 1. - Charles R Greathouse IV, May 13 2019
The other positive root of the minimal polynomial is A019952. - R. J. Mathar, Sep 06 2025

			0.3249196962329063261558714122151344649549034715214751003078047191...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Exact trigonometric constants
Index entries for algebraic numbers, degree 4.

Cf. A001622, A019827 (sin(Pi/10)), A019881 (cos(Pi/10)).

RealDigits[Tan[18 Degree],10,120][[1]] (* Harvey P. Dale, Mar 07 2012 *)

tan(Pi/10) \\ Michel Marcus, Jan 08 2018

polrootsreal(5*x^4-10*x^2+1)[3] \\ Charles R Greathouse IV, Feb 04 2025

Equals A019827/A019881 = 1/A019970 = 1/sqrt(5+2*sqrt(5)). - R. J. Mathar, Jul 26 2010
Equals tan((phi - 1)/sqrt(2 + phi)) = (1/5)*(sqrt(3 - phi))^3 = (3 - phi)*sqrt(3 - phi)/5 = sqrt(7 - 4*phi)/(2*phi - 1), with phi from A001622. - Wolfdieter Lang, Jan 08 2018
Equals Product_{k>=0} ((5*k + 1)/(5*k + 4))^(-1)^(k) = Product_{k>=0} A090771(k)/A090773(k). - Antonio Graciá Llorente, Mar 24 2024
Equals A019845/(1+A019863). - R. J. Mathar, Sep 06 2025

A090772 Numbers that are congruent to {2, 8} mod 10.

Original entry on oeis.org

2, 8, 12, 18, 22, 28, 32, 38, 42, 48, 52, 58, 62, 68, 72, 78, 82, 88, 92, 98, 102, 108, 112, 118, 122, 128, 132, 138, 142, 148, 152, 158, 162, 168, 172, 178, 182, 188, 192, 198, 202, 208, 212, 218, 222, 228, 232, 238, 242, 248, 252, 258, 262, 268, 272, 278, 282

Offset: 1

Giovanni Teofilatto, Feb 07 2004

Their square ends in the digit 4. - Kausthub Gudipati, Sep 08 2011

10*a(n) = 20, 80, 120, 180, 220, ... are the only numbers written in French ending in "vingt(s)". - Paul Curtz, Aug 02 2018

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A019952, A047209, A090298, A179290, A226294, A317633.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(2*x*(1+3*x+x^2)/((1+x)*(1-x)^2))); // G. C. Greubel, Aug 08 2018

Union@ Flatten@ Outer[Plus, {2, 8}, 10 Range[0, 28]] (* or *)
CoefficientList[Series[2 (1 + 3x + x^2)/((1 + x) (1 - x)^2), {x, 0, 57}], x] (* Michael De Vlieger, Aug 02 2018 *)
LinearRecurrence[{1, 1, -1}, {2, 8, 12}, 61] (* Robert G. Wilson v, Aug 08 2018 *)

is(n) = #setintersect([2, 8], [n%10]) > 0 \\ Felix Fröhlich, Aug 02 2018

Vec(2*x*(1+3*x+x^2)/((1+x)*(1-x)^2) + O(x^60)) \\ Felix Fröhlich, Aug 02 2018

a(n) = 2 * A047209(n).
a(n) = 10*n - a(n-1) - 10 (with a(1)=2). - Vincenzo Librandi, Nov 16 2010
G.f.: 2*x*(1+3*x+x^2)/((1+x)*(1-x)^2). - Bruno Berselli, Sep 08 2011
a(1) = 2. For n > 1, a(n) = a(n-1) + A226294(n). - Felix Fröhlich, Aug 02 2018
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(1+2/sqrt(5))*Pi/10. - Amiram Eldar, Dec 28 2021
E.g.f.: 2 + ((10*x - 5)*exp(x) + exp(-x))/2. - David Lovler, Sep 03 2022
From Amiram Eldar, Nov 23 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = tan(3*Pi/10) (A019952).
Product_{n>=1} (1 + (-1)^n/a(n)) = cosec(2*Pi/5)/2 (= A179290 / 2). (End)

Edited and extended by Ray Chandler, Feb 10 2004

A344171 Decimal expansion of 12*sqrt(5).

Original entry on oeis.org

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

Offset: 2

Wesley Ivan Hurt, May 10 2021

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

			26.83281572999747635691008...

Wikipedia, Rhombic triacontahedron

Cf. A344172 (rhombic triacontahedron volume).
Cf. A344212 (rhombic triacontahedron midradius).
Cf. A019952 (rhombic triacontahedron radius of inscribed sphere).

RealDigits[12 Sqrt[5], 10, 100][[1]] // Flatten

cp's OEIS Frontend

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

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A033571 a(n) = (2*n + 1)*(5*n + 1).

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A344212 Decimal expansion of 1 + 1/sqrt(5).

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A019934 Decimal expansion of tangent of 36 degrees.

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A200135 Decimal expansion of the negated value of the digamma function at 1/5.

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A237603 Decimal expansion of the inscribed sphere radius in a regular dodecahedron with unit edge.

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A033571 a(n) = (2n + 1)(5*n + 1).