A090773 -id:A090773 - cp's OEIS frontend

A179290 Decimal expansion of length of edge of a regular icosahedron with radius of circumscribed sphere = 1.

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 09 2010

Regular icosahedron: A three-dimensional figure with 20 congruent equilateral triangle faces, 12 vertices, and 30 edges.
Shorter diagonal of golden rhombus with unit edge length. - Eric W. Weisstein, Dec 11 2018
The length of the shorter side of a golden rectangle inscribed in a unit circle. - Michal Paulovic, Sep 01 2022
The side length of a square inscribed within a golden ellipse with a unit semi-major axis. - Amiram Eldar, Oct 02 2022
(10/3)*(this constant)=3.504874080794224... is the volume of the polyhedron with 32 edges with conjectured maximum volume inscribed in a sphere of radius 1. It has 60 congruent triangular faces and the symmetry group of the regular icosahedron. See Pfoertner links for visualizations. - Hugo Pfoertner, Aug 02 2025

			1.051462224238267212051338169695753214570995864486683563057871046482422...

Muniru A Asiru, Table of n, a(n) for n = 1..1000
J. Brandts, S. Korotov, M. Krizek, and J. Solc, On nonobtuse simplicial partitions, Siam Rev. 51 (2) (2009) 317-335.
Dr. Math, Regular Polyhedra: Formulas.
Hugo Pfoertner, Visualization of the 32-edge polyhedron with conjecturally maximum volume, (2021).
Hugo Pfoertner, Number of edges incident with the vertices of the 32-edge polyhedron, video (2021).
Eric Weisstein's World of Mathematics, Golden Rhombus.
Eric Weisstein's World of Mathematics, Icosahedron.
Wikipedia, Icosahedron.
Index entries for algebraic numbers, degree 4.

Cf. A010527, A019881, A090773, A102208.

Cf. A179290 (longer golden rhombus diagonal).

evalf[120](csc(2*Pi/5)); # Muniru A Asiru, Dec 11 2018

RealDigits[Csc[2 Pi/5], 10, 110][[1]] (* Eric W. Weisstein, Dec 11 2018 *)

sqrt(50-10*sqrt(5))/5 \\ Charles R Greathouse IV, Jan 22 2024

from decimal import *
getcontext().prec = 110
c = Decimal.sqrt(2 - 2 / Decimal.sqrt(Decimal(5)))
print([int(i) for i in str(c) if i != '.'])
# Karl V. Keller, Jr., Jul 10 2020

Equals sqrt(50-10*sqrt(5))/5.
Equals csc(2*Pi/5). - Eric W. Weisstein, Dec 11 2018
Equals 1/Im(e^(3*i*Pi/5)) = 1/Im(e^(3*i*Pi/5) - 1) = sqrt(2 - 2/sqrt(5)). - Karl V. Keller, Jr., Jun 11 2020
Equals 1/A019881. - R. J. Mathar, Jan 17 2021
From Antonio Graciá Llorente, Mar 15 2024: (Start)
Equals Product_{k >= 1} ((10*k - 1)*(10*k + 1))/((10*k - 2)*(10*k + 2)).
Equals Product_{k >= 1} 1/(1 - 1/(25*(2*k - 1)^2)). (End)
Equals Product_{k>=1} (1 - (-1)^k/A090773(k)). - Amiram Eldar, Nov 23 2024
A root of 5*x^4 - 20*x^2 + 16=0 (see A121570). - R. J. Mathar, Aug 29 2025

Partially rewritten by Charles R Greathouse IV, Feb 02 2011

A090298 Permutation of natural numbers generated by 5-row array shown below.

Original entry on oeis.org

1, 9, 2, 11, 8, 3, 19, 12, 7, 4, 21, 18, 13, 6, 5, 29, 22, 17, 14, 10, 31, 28, 23, 16, 15, 39, 32, 27, 24, 20, 41, 38, 33, 26, 25, 49, 42, 37, 34, 30, 51, 48, 43, 36, 35, 59, 52, 47, 44, 40, 61, 58, 53, 46, 45, 69, 62, 57, 54, 50, 71, 68, 63, 56, 55, 79, 72, 67, 64, 60, 81, 78

Offset: 1

Giovanni Teofilatto, Jan 25 2004

9 11 19 21 29 31 39... (A090771)
8 12 18 22 28 32 38... (A090772)
7 13 17 23 27 33 37... (A063226)
6 14 16 24 26 34 36... (A090773)
10 15 20 25 30 35 40... (A008587, excluding initial term)
-----------------------------------------------------------
For such arrays A_k, here A_5, see a W. Lang comment on A113807, the A_7 case. However, in order to obtain A_5 one should take the last row as the first one after adding a 0 in front (thus getting a permutation of the nonnegative integers). - Wolfdieter Lang, Feb 02 2012

Cf. A088520, A090243, A092260, A113807.

More terms from Ray Chandler, Feb 01 2004

A300074 Decimal expansion of 1/(2*sin(Pi/5)) = A121570/2.

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Mar 01 2018

This is the reciprocal of A182007, and one half of A121570.
This is the ratio of the radius r of the circumscribing circle of a regular pentagon and its side length s: r/s = 1/(2*sin(Pi/5)).
A quartic number of denominator 5 and minimal polynomial 5x^4 - 5x^2 + 1. - Charles R Greathouse IV, Mar 04 2018
Appears at Schur decomposition of A=[1 2; 2 3]. - Donghwi Park, Jun 20 2018

			r/s = 0.850650808352039932181540497063011072240401403764816881836740242377...
2*r/s = A121570.

Muniru A Asiru, Table of n, a(n) for n = 0..2000
Wikipedia, Schur decomposition.
Index entries for algebraic numbers, degree 4.

Cf. A001622, A090773, A121570, A182007, A195693, A195723.

RealDigits[1/(2 Sin[Pi/5]), 10, 111][[1]] (* Robert G. Wilson v, Jul 15 2018 *)

1/(2*sin(Pi/5)) \\ Charles R Greathouse IV, Mar 04 2018

sqrt((5+sqrt(5))/10) \\ Charles R Greathouse IV, Mar 04 2018

r/s = 1/A182007 = A121570/2 = (2*phi - 1)*sqrt(2 + phi)/5, with the golden ratio phi = (1 + sqrt(5))/2 = A001622.
From Amiram Eldar, Feb 08 2022: (Start)
Equals cos(arccot(phi)) = cos(arctan(1/phi)) = cos(A195693).
Equals sin(arctan(phi)) = sin(arccot(1/phi)) = sin(A195723). (End)
Equals Product_{k>=1} (1 + (-1)^k/A090773(k)). - Amiram Eldar, Nov 23 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

			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

A273373 Squares ending in digit 6.

Original entry on oeis.org

16, 36, 196, 256, 576, 676, 1156, 1296, 1936, 2116, 2916, 3136, 4096, 4356, 5476, 5776, 7056, 7396, 8836, 9216, 10816, 11236, 12996, 13456, 15376, 15876, 17956, 18496, 20736, 21316, 23716, 24336, 26896, 27556, 30276, 30976, 33856, 34596, 37636, 38416, 41616

Offset: 1

Vincenzo Librandi, May 21 2016

These are the only squares whose second last digit is odd. This implies that the only squares whose last two digits are the same are those ending with 0 or 4; those ending with 1, 5, and 9 are paired with even second last digits. - Waldemar Puszkarz, May 24 2016

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000290, A047221, A090773.
Cf. A017341 (numbers ending in 6), A017343 (cubes ending in 6).
Cf. squares with last digit k: A017270 (k=0), A273372 (k=1), A273375 (k=4), A017330 (k=5), this sequence (k=6), A273374 (k=9).

/* By definition: */ [n^2: n in [0..200] | Modexp(n,2,10) eq 6];

[(10*n - 3*(-1)^n - 5)^2/4: n in [1..50]];

seq(seq((10*i+j)^2,j=[4,6]),i=0..20); # Robert Israel, May 24 2016

Table[(10 n - 3 (-1)^n - 5)^2/4, {n, 1, 50}]
CoefficientList[Series[4 (4 + 5 x + 32 x^2 + 5 x^3 + 4 x^4) / ((1 + x)^2 (1 - x)^3), {x, 0, 50}], x]
Select[Range[250]^2,Mod[#,10]==6&] (* Harvey P. Dale, May 31 2020 *)

G.f.: 4*x*(4 + 5*x + 32*x^2 + 5*x^3 + 4*x^4)/((1 + x)^2*(1 - x)^3).
a(n) = 4*A047221(n)^2 = (10*n - 3*(-1)^n - 5)^2/4.
a(n) = A090773(n)^2. - Michel Marcus, May 25 2016
Sum_{n>=1} 1/a(n) = 2*Pi^2/(25*(5+sqrt(5))). - Amiram Eldar, Feb 16 2023

Corrected and extended by Bruno Berselli, May 23 2016

A284477 Pairs of integers (x, y), such that x^2 + 1 and y^2 + 1, 1 < y < x, have the same distinct prime factors.

Original entry on oeis.org

7, 3, 18, 8, 117, 43, 239, 5, 378, 132, 843, 377, 2207, 987, 2943, 73, 4443, 53, 4662, 1568, 6072, 5118, 8307, 743, 8708, 2112, 9872, 2738, 31561, 4929, 103682, 46368, 271443, 121393, 853932, 76378, 1021693, 91383, 3539232, 41218, 3699356, 473654

Offset: 1

Michel Lagneau, Mar 27 2017

nonn

The sequence appears to thin out quite abruptly; however, by solving the Diophantine equation x^2 + 1 = p (y^2 + 1) for a suitable prime p and selecting the solutions (x, y) for which p divides y^2 + 1, it is easy to generate larger pairs, such as (423222288438379883442890018716361, 66096216900526495715353522199871). - Giovanni Resta, Mar 27 2017
A very interesting property: the sequence contains a subsequence of pairs (Lucas numbers L(i), Fibonacci numbers F(i)) for i = 4, 6, 14, 16, 24, 36, ... These pairs are (L(4), F(4)), (L(6), F(6)), (L(14), F(14)), (L(16), F(16)), (L(24), F(24)), (L(26), F(26)), (L(34), F(34)), ... = (7, 3), (18, 8), (843, 377), (2207, 987), (103682, 46368), (271443, 121393), (12752043, 5702887), ... It seems that {i} = A090773(n) (numbers that are congruent to {4, 6} mod 10). - Michel Lagneau, Mar 28 2017
This is because L(i)^2+1 = 5*(F(i)^2+1) for even i, and 5 | F(i)^2 + 1 for i== 3,4,6,7 (mod 10).  In fact (L(i), F(i)) for i in A090773 are the solutions of the generalized Pell equation x^2 + 1 = 5 (y^2 + 1) for which 5 | y^2 + 1. - Robert Israel, Apr 10 2017

			The pair (843, 377) is in the sequence because the prime factors of 843^2 + 1 and 377^2 + 1 are 2, 5, 61 and 233.

Giovanni Resta, Table of n, a(n) for n = 1..68 (terms with x < 1.5*10^8)

Cf. A002496, A089122, A128428.

A:= NULL:
for x from 2 to 10^5 do
  P:= numtheory:-factorset(x^2+1);
  if not assigned(R[P]) then R[P]:= x
  else A:= A, op(map(t -> (x,t), [R[P]]));
       R[P]:= R[P],x
  fi
od:
A; # Robert Israel, Apr 10 2017

d[n_] := First /@ FactorInteger[n]; Flatten@ Reap[ Do[ dx = d[x^2+1]; Do[ If[ dx == d[y^2+1], Sow[{x, y}]], {y, x-1}], {x, 1, 10^4}]][[2, 1]]

upto(n) = {my(l = List(), res=List()); for(i=1, n, f = factor(i^2+1)[, 1]; listput(l, [f, i])); listsort(l); for(i=1, n-1, if(l[i][1]==l[i+1][1], listput(res, [l[i+1][2], l[i][2]]))); listsort(res); res} \\ David A. Corneth, Mar 28 2017

a(29)-a(34) from Giovanni Resta, Mar 27 2017

a(35)-a(42) from David A. Corneth, Mar 28 2017

cp's OEIS Frontend

A179290 Decimal expansion of length of edge of a regular icosahedron with radius of circumscribed sphere = 1.

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A090298 Permutation of natural numbers generated by 5-row array shown below.

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A300074 Decimal expansion of 1/(2*sin(Pi/5)) = A121570/2.

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

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A273373 Squares ending in digit 6.

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A284477 Pairs of integers (x, y), such that x^2 + 1 and y^2 + 1, 1 < y < x, have the same distinct prime factors.

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Maple

Mathematica

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