A296182 -id:A296182 - cp's OEIS frontend

A019863 Decimal expansion of sin(3*Pi/10) (sine of 54 degrees, or cosine of 36 degrees).

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

Offset: 0

N. J. A. Sloane

Midsphere radius of regular icosahedron with unit edges.
Also half of the golden ratio (A001622). - Stanislav Sykora, Jan 30 2014
Andris Ambainis (see Aaronson link) observes that combining the results of Barak-Hardt-Haviv-Rao with Dinur-Steurer yields the maximal probability of winning n parallel repetitions of a classical CHSH game (see A201488) asymptotic to this constant to the power of n, an improvement on the naive probability of (3/4)^n. (All the random bits are received upfront but the players cannot communicate or share an entangled state.) - Charles R Greathouse IV, May 15 2014
This is the height h of the isosceles triangle in a regular pentagon, in length units of the circumscribing radius, formed by a side as base and two adjacent radii. h = sin(3*Pi/10) = cos(Pi/5) (radius 1 unit). - Wolfdieter Lang, Jan 08 2018
Also the limiting value(L) of "r" which is abscissa of the vertex of the parabola  F(n)*x^2 - F(n+1)*x + F(n + 2)(where F(n)=A000045(n) are the Fibonacci numbers and n>0). - Burak Muslu, Feb 24 2021

			0.80901699437494742410229341718281905886015458990288143106772431135263...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Scott Aaronson, The NEW Ten Most Annoying Questions in Quantum Computing (2014).
Boaz Barak, Moritz Hardt, Ishay Haviv, and Anup Rao, Rounding Parallel Repetitions of Unique Games (2008).
Irit Dinur and David Steurer, Analytical approach to parallel repetition, arXiv:1305.1979 [cs.CC], 2013-2014.
Michael Penn, a golden value of cosine., YouTube video, 2021.
Wikipedia, Exact trigonometric constants.
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 2.

Cf. A001611, A001622, A019827, A134972, A134944, A134946, A187426, A296182.

Platonic solids midradii: A020765 (tetrahedron), A020761 (octahedron), A010503 (cube), A239798 (dodecahedron).

convert(sin(3*Pi/10),radical); # W. Edwin Clark, May 24 2023
Digits:=100; evalf((1+sqrt(5))/4); # Wesley Ivan Hurt, Mar 27 2014

RealDigits[(1 + Sqrt[5])/4, 10, 111] (* Robert G. Wilson v *)
RealDigits[Sin[54 Degree],10,120][[1]] (* Harvey P. Dale, Apr 21 2018 *)

(1+sqrt(5))/4 \\ Charles R Greathouse IV, Jan 16 2012

Equals (1+sqrt(5))/4 = cos(Pi/5) = sin(3*Pi/10). - R. J. Mathar, Jun 18 2006
Equals 2F1(4/5,1/5;1/2;3/4) / 2 = A019827 + 1/2. - R. J. Mathar, Oct 27 2008
Equals A001622 / 2. - Stanislav Sykora, Jan 30 2014
phi / 2 = (i^(2/5) + i^(-2/5)) / 2 = i^(2/5) - (sin(Pi/5))*i = i^(-2/5) + (sin(Pi/5))*i = i^(2/5) - (cos(3*Pi/10))*i = i^(-2/5) + (cos(3*Pi/10))*i. - Jaroslav Krizek, Feb 03 2014
Equals 1/A134972. - R. J. Mathar, Jan 17 2021
Equals 2*A019836*A019872. - R. J. Mathar, Jan 17 2021
Equals (A094214 + 1)/2 or 1/(2*A094214). - Burak Muslu, Feb 24 2021
Equals hypergeom([-2/5, -3/5], [6/5], -1) = hypergeom([-1/5, 3/5], [6/5], 1) = hypergeom([1/5, -3/5], [4/5], 1). - Peter Bala, Mar 04 2022
Equals Product_{k>=1} (1 - (-1)^k/A001611(k)). - Amiram Eldar, Nov 28 2024
Equals 2*A134944 = 3*A134946 = A187426-11/10 = A296182-1. - Hugo Pfoertner, Nov 28 2024
Equals A134945/4. Root of 4*x^2-2*x-1=0. - R. J. Mathar, Aug 29 2025

A179133 Denominators of A178381(4n+3)/A178381(4n+2).

Original entry on oeis.org

2, 4, 5, 26, 68, 89, 466, 1220, 1597, 8362, 21892, 28657, 150050, 392836, 514229, 2692538, 7049156, 9227465, 48315634, 126491972, 165580141, 866988874, 2269806340, 2971215073, 15557484098, 40730022148, 53316291173, 279167724890

Offset: 0

Johannes W. Meijer, Jul 01 2010

For the numerators see A128052.

Index entries for linear recurrences with constant coefficients, signature (0,0,18,0,0,-1).

Cf. A000045, A128052, A153727, A178381, A179131, A179132, A179133, A179134, A296182.

with(GraphTheory): nmax:=120; P:=9: G:=PathGraph(P): A:= AdjacencyMatrix(G): for n from 0 to nmax do B(n):=A^n; A178381(n):=add(B(n)[1,k],k=1..P); od: for n from 0 to nmax-1 do a(n):= denom(A178381(4*n+3)/A178381(4*n+2)) od: seq(a(n),n=0..nmax/4-1);

Flatten[Table[{2*Fibonacci[6 n + 1], 2*Fibonacci[6 n + 3],
Fibonacci[6 n + 5]}, {n, 0, 10}]] (* Greg Dresden, Oct 16 2021 *)
LinearRecurrence[{0,0,18,0,0,-1},{2,4,5,26,68,89},30] (* Harvey P. Dale, Oct 08 2024 *)

a(n) = A179134(n)*A153727(n+1)/2.
Lim_{n->infinity} A128052(n+1)/A179133(n) = 1+cos(Pi/5) = A296182.
From Colin Barker, Jun 27 2013: (Start)
G.f.: -(x^5+4*x^4+10*x^3-5*x^2-4*x-2)/((x^2-3*x+1)*(x^4+3*x^3+8*x^2+3*x+1)).
a(n) = 18*a(n-3)-a(n-6). (End)
From Greg Dresden, Oct 16 2021: (Start)
a(3*n) = 2*Fibonacci(6*n+1),
a(3*n+1) = 2*Fibonacci(6*n+3),
a(3*n+2) = Fibonacci(6*n+5). (End)

A179131 Numerators of A178381(4n+1)/A178381(4n).

Original entry on oeis.org

1, 5, 25, 65, 85, 445, 1165, 1525, 7985, 20905, 27365, 143285, 375125, 491045, 2571145, 6731345, 8811445, 46137325, 120789085, 158114965, 827900705, 2167472185, 2837257925, 14856075365, 38893710245, 50912527685, 266581455865

Offset: 0

Johannes W. Meijer, Jul 01 2010

For the denominators see A179132.

Index entries for linear recurrences with constant coefficients, signature (0,0,18,0,0,-1).

Cf. A128052, A179133, A179132.

with(GraphTheory): nmax:=116; P:=9: G:=PathGraph(P): A:= AdjacencyMatrix(G): for n from 0 to nmax do B(n):=A^n; A178381(n):=add(B(n)[1,k],k=1..P); od: for n from 0 to nmax-1 do a(n):= numer(A178381(4*n+1)/A178381(4*n)) od: seq(a(n),n=0..nmax/4-1);

a(n) = 5*A167808(2*n+1) for n>=1.
Limit(A179131(n)/A179132(n), n =infinity) = 1+cos(Pi/5) = A296182.
a(n) = 18*a(n-3)-a(n-6) for n>6. G.f.: -(4*x^6+5*x^5+5*x^4-47*x^3-25*x^2-5*x-1) / ((x^2-3*x+1)*(x^4+3*x^3+8*x^2+3*x+1)). - Colin Barker, Jun 27 2013

A179132 Denominators of A178381(4n+1)/A178381(4n).

Original entry on oeis.org

1, 3, 14, 36, 47, 246, 644, 843, 4414, 11556, 15127, 79206, 207364, 271443, 1421294, 3720996, 4870847, 25504086, 66770564, 87403803, 457652254, 1198149156, 1568397607, 8212236486, 21499914244, 28143753123, 147362604494

Offset: 0

Johannes W. Meijer, Jul 01 2010

For the numerators see A179131.

Index entries for linear recurrences with constant coefficients, signature (0,0,18,0,0,-1).

Cf. A128052 and A179133.

with(GraphTheory): nmax:=116; P:=9: G:=PathGraph(P): A:= AdjacencyMatrix(G): for n from 0 to nmax do B(n):=A^n; A178381(n):=add(B(n)[1,k],k=1..P); od: for n from 0 to nmax-1 do a(n):= denom(A178381(4*n+1)/A178381(4*n)) od: seq(a(n),n=0..nmax/4-1);

LinearRecurrence[{0,0,18,0,0,-1},{1,3,14,36,47,246,644},30] (* Harvey P. Dale, Jun 11 2022 *)

a(n) = A069705(n-1)*A128052(n) for n>=1.
Limit(A179131(n)/A179132(n), n =infinity) = 1+cos(Pi/5) = A296182.
a(n) = 18*a(n-3)-a(n-6) for n>6. G.f.: -(3*x^6+6*x^5+7*x^4-18*x^3-14*x^2-3*x-1) / ((x^2-3*x+1)*(x^4+3*x^3+8*x^2+3*x+1)). - Colin Barker, Jun 27 2013

A322159 Decimal expansion of 1 - 1/sqrt(5).

Original entry on oeis.org

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

Offset: 0

Tristan Cam, Nov 29 2018

Continued fraction: [0;1,1,4,4,4...].

Least root of the polynomial: 5x^2 - 10x + 4.

			0.552786404500042060718165266253744752911876328077...

Cf. A001622, A002163, A020762, A081010, A244847, A296182.

evalf[110](1-1/sqrt(5)); # Muniru A Asiru, Dec 01 2018

RealDigits[1-1/Sqrt[5], 10, 100][[1]] (* Amiram Eldar, Nov 29 2018 *)

Equals 1 - 1/A002163.
Equals 1/(1 - cos(4*Pi/5)) = (1/2)*csc(2*Pi/5)^2.
Also equal to 2/(phi*sqrt(5)) = 2/(A001622*A002163).
Equals 1 - A020762. - Andrew Howroyd, Nov 30 2018
From Amiram Eldar, Nov 28 2024: (Start)
Equals 2*A244847 = 1/A296182.
Equals Product_{k>=0} (1 - 1/A081010(k)). (End)

A081003 a(n) = Fibonacci(4n+1) + 1, or Fibonacci(2n+1)*Lucas(2n).

Original entry on oeis.org

2, 6, 35, 234, 1598, 10947, 75026, 514230, 3524579, 24157818, 165580142, 1134903171, 7778742050, 53316291174, 365435296163, 2504730781962, 17167680177566, 117669030460995, 806515533049394, 5527939700884758, 37889062373143907, 259695496911122586

Offset: 0

R. K. Guy, Mar 01 2003

Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75

Nathaniel Johnston, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Cf. A000045 (Fibonacci numbers), A000032 (Lucas numbers), A296182.

List([0..30], n-> Fibonacci(4*n+1)+1); # G. C. Greubel, Jul 15 2019

[Fibonacci(4*n+1) +1: n in [0..30]]; // Vincenzo Librandi, Apr 15 2011

with(combinat): for n from 0 to 30 do printf(`%d,`,fibonacci(4*n+1)+1) od:  # James Sellers, Mar 03 2003

Fibonacci[4*Range[0,30]+1]+1 (* or *) LinearRecurrence[{8,-8,1}, {2,6,35}, 30] (* Harvey P. Dale, Jul 20 2011 *)

vector(30, n, n--; fibonacci(4*n+1)+1) \\ G. C. Greubel, Jul 15 2019

[fibonacci(4*n+1)+1 for n in (0..30)] # G. C. Greubel, Jul 15 2019

a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3).
G.f.: (2-10*x+3*x^2)/((1-x)*(1-7*x+x^2)). - Colin Barker, Jun 24 2012
Product_{n>=0} (1 + 1/a(n)) = (2 + phi)/2 (A296182). - Amiram Eldar, Nov 28 2024

More terms from James Sellers, Mar 03 2003

cp's OEIS Frontend

A019863 Decimal expansion of sin(3*Pi/10) (sine of 54 degrees, or cosine of 36 degrees).

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A179133 Denominators of A178381(4*n+3)/A178381(4*n+2).

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A179131 Numerators of A178381(4*n+1)/A178381(4*n).

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A179132 Denominators of A178381(4*n+1)/A178381(4*n).

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A322159 Decimal expansion of 1 - 1/sqrt(5).

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A081003 a(n) = Fibonacci(4n+1) + 1, or Fibonacci(2n+1)*Lucas(2n).

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A179133 Denominators of A178381(4n+3)/A178381(4n+2).

A179131 Numerators of A178381(4n+1)/A178381(4n).

A179132 Denominators of A178381(4n+1)/A178381(4n).