A187798 -id:A187798 - cp's OEIS frontend

A019827 Decimal expansion of sin(Pi/10) (angle of 18 degrees).

3, 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

Decimal expansion of cos(2*Pi/5) (angle of 72 degrees).
Also the imaginary part of i^(1/5). - Stanislav Sykora, Apr 25 2012
One of the two roots of 4x^2 + 2x - 1 (the other is the sine of 54 degrees times -1 = -A019863). - Alonso del Arte, Apr 25 2015
This is the height h of the isosceles triangle in a regular pentagon inscribed in a unit circle, formed by a diagonal as base and two adjacent radii. h = cos(2*Pi/5) = sin(Pi/10). - Wolfdieter Lang, Jan 08 2018
Quadratic number of denominator 2 and minimal polynomial 4x^2 + 2x - 1. - Charles R Greathouse IV, May 13 2019
Largest superstable width of the logistic map (see Finch). - Stefano Spezia, Nov 23 2024

			0.30901699437494742410229341718281905886015458990288143106772431135263...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 1.9 and 8.19, pp. 66, 535.

Zak Seidov, Table of n, a(n) for n = 0..999
Hideyuki Ohtsuka, Problem B-1237, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 56, No. 4 (2018), p. 366; A Telescoping Product, Solution to Problem B-1237 by Steve Edwards, ibid., Vol. 57, No. 4 (2019), pp. 369-370.
Wikipedia, Exact trigonometric constants.
Index entries for algebraic numbers, degree 2.

Cf. A001622, A019845, A019863, A055588, A094214, A134945, A187798, A341332, A377697.

RealDigits[Sin[18 Degree], 10, 108][[1]] (* Alonso del Arte, Apr 20 2015 *)

sin(Pi/10) \\ Charles R Greathouse IV, Feb 03 2015

polrootsreal(4*x^2 + 2*x - 1)[2] \\ Charles R Greathouse IV, Feb 03 2015

Equals (sqrt(5) - 1)/4 = (phi - 1)/2 = 1/(2*phi), with phi from A001622.
Equals 1/(1 + sqrt(5)). - Omar E. Pol, Nov 15 2007
Equals 1/A134945. - R. J. Mathar, Jan 17 2021
Equals 2*A019818*A019890. - R. J. Mathar, Jan 17 2021
Equals Product_{k>=1} 1 - 1/(phi + phi^k), where phi is the golden ratio (A001622) (Ohtsuka, 2018). - Amiram Eldar, Dec 02 2021
Equals Product_{k>=1} (1 - 1/A055588(k)). - Amiram Eldar, Nov 28 2024
Equals A094214/2 = 1-A187798 = A341332/Pi = (A377697-2)/3. - Hugo Pfoertner, Nov 28 2024
This^2 + A019881^2 = 1. - R. J. Mathar, Aug 31 2025

A019845 Decimal expansion of sine of 36 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

This sequence is also decimal expansion of cosine of 54 degrees. - Mohammad K. Azarian, Jun 29 2013
The ratio of side to longer diagonal for any golden rhombus (see A019881). - Rick L. Shepherd, Apr 10 2017
Perimeter length of a regular pentagon with circumscribed unit circle. - R. J. Mathar, Aug 24 2023

			sin 36 degrees = 0.587785252292473129168705954639...

Zak Seidov, Table of n, a(n) for n = 0..999
Wikipedia, Exact trigonometric constants
Index entries for algebraic numbers, degree 4

Cf. A019827 (sine of 18 degrees), A019881 (sine of 72 degrees), A001622 (golden ratio phi). A182007.

RealDigits[Sin[Pi/5], 10, 100][[1]] (* Alonso del Arte, Sep 19 2017 *)
RealDigits[Sin[36 Degree],10,120][[1]] (* Harvey P. Dale, Aug 14 2018 *)

sin(Pi/5) \\ Michel Marcus, Apr 25 2015

cos(3*Pi/10) \\ Rick L. Shepherd, Apr 10 2017

real(I^(3/5)) \\ Rick L. Shepherd, Apr 10 2017

sin 36 degrees = sin Pi/5 radians = sqrt((1/8)(5 - sqrt(5))) = sqrt(A187798/2).
Equals A019881/A001622. - Rick L. Shepherd, Apr 10 2017
This constant is (1/2)*A182007. - Wolfdieter Lang, May 08 2018
Equals 2*A019827*A019881. - R. J. Mathar, Jan 17 2021
Equals 5*A182007. - R. J. Mathar, Aug 24 2023
Equals cos(3*Pi/10). - R. J. Mathar, Aug 29 2025
Root of 16*x^4-20*x^2+5=0. Other 2 roots are +- A019881. - R. J. Mathar, Aug 29 2025
This^2+A019863^2=1. - R. J. Mathar, Aug 31 2025

A094874 Decimal expansion of (5-sqrt(5))/2.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jun 14 2004

Also the limiting ratio of Lucas(n)/Fibonacci(n+1), or Fibonacci(n-1)/Fibonacci(n+1) + 1. - Alexander Adamchuk, Oct 10 2007

			1.38196601125010515179541316563436188...

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Paul Cooijmans, Odds.
Yiyan Ni, Myron Hlynka, and Percy H. Brill, Urn Models and Fibonacci Series, arXiv:1806.09150 [math.CO], 2018. See (9) p. 7.
Jonathan Sondow, Evaluation of Tachiya's algebraic infinite products involving Fibonacci and Lucas numbers, arXiv:1106.4246 [math.NT], 2011; Diophantine Analysis and Related Fields 2011 - AIP Conference Proceedings, vol. 1385, pp. 97-100.
Index entries for algebraic numbers, degree 2.

Equals A079585-1.

Cf. A000032, A000045, A081012, A182007, A187426, A187798, A192223, A242671, A244847.

RealDigits[5/2 - Sqrt[5]/2, 10, 100][[1]] (* Alonso del Arte, Jun 26 2018 *)

(5-sqrt(5))/2 \\ Charles R Greathouse IV, Jun 26 2011

Equals (2-phi)*(2+phi) = 2 - 1/phi = 3 - phi = (5-sqrt(5))/2 = (2*sin(Pi/5))^2, where phi is the golden ratio (A001622).
Equals Product_{n > 0} (1 + 1/A192223(n)). - Charles R Greathouse IV, Jun 26 2011
Equals 1 + Sum_{k >= 2} (-1)^k/(Fibonacci(k)*Fibonacci(k+1)). See Ni et al. - Michel Marcus, Jun 26 2018; corrected by Michel Marcus, Mar 11 2024
Equals Sum_{k>=0} binomial(2*k,k)/((k+1) * 5^k). - Amiram Eldar, Aug 03 2020
From Amiram Eldar, Nov 28 2024: (Start)
Equals 5*A244847 = 2*A187798 = 1/A242671 = A182007^2 = sqrt(A187426).
Equals Product_{k>=1} (1 + 1/A081012(k)). (End)

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)

A187426 Decimal expansion of (3-phi)^2 = 10 - 5*phi where phi is the golden ratio.

Original entry on oeis.org

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

Offset: 1

Joost Gielen, Aug 30 2013

This is an integer in Q(sqrt(5)). - Wolfdieter Lang, Jan 07 2018

			1.909830...

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Index entries for algebraic numbers, degree 2.

Cf. A226765.

Apart from the first digit the same as A187798.

RealDigits[(3-GoldenRatio)^2,10,120][[1]] (* Harvey P. Dale, Jun 30 2017 *)

(15-5*sqrt(5))/2 \\ Charles R Greathouse IV, Aug 31 2013

polrootsreal(x^2-15*x+25)[1] \\ Charles R Greathouse IV, Feb 04 2025

(3-phi)^2 = 5/phi^2 = 10 - 5*phi.
Smaller root of x^2 - 15x + 25 = 0.
Equals 10*A187798-5. - R. J. Mathar, Feb 08 2023

Corrected and extended by Charles R Greathouse IV, Aug 31 2013

A187799 Decimal expansion of 20/phi^2, where phi is the golden ratio. Also (with a different offset), decimal expansion of 3 - sqrt(5).

Original entry on oeis.org

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

Offset: 1

Joost Gielen, Aug 30 2013

			20/phi^2 = 7.6393202250021030359082633...
3 - sqrt(5) = 0.76393202250021030359082633... (with offset 0).

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Mohammad K. Azarian, The Value of a Series of Reciprocal Fibonacci Numbers, Problem B-1133, Fibonacci Quarterly, Vol. 51, No. 3, August 2013, p. 275; Solution published in Vol. 52, No. 3, August 2014, pp. 277-278.

Cf. A187426, A187798, A094874.

5*(Sqrt(5)-1)^2; // Vincenzo Librandi, Feb 24 2015

First@ RealDigits[N[5*(Sqrt[5] - 1)^2, 111]] (* Michael De Vlieger, Feb 25 2015 *)

5*(sqrt(5)-1)^2 \\ Charles R Greathouse IV, Aug 31 2013

10*(3 - sqrt(5)) = 30 - 10*sqrt(5) = (5 - sqrt(5))^2 = 20/phi^2.

2 * Sum_{i > 1} (-1)^i/(F(i)F(i + 1)) = 3 - sqrt(5), where F(i) is the i-th Fibonacci number. This formula comes from John D. Watson, Jr.'s solution to Azarian's Problem B-1133 in the Fibonacci Quarterly. Azarian originally posed the problem as an infinite alternating sum explicitly written out for the first dozen terms or so. See the Azarian links above. - Alonso del Arte, Aug 25 2016

Extended by Charles R Greathouse IV, Aug 31 2013

A226765 Decimal expansion of (13-5*sqrt(5))/2.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Jul 21 2013

(13 - 5*sqrt(5))/2 = lim_{n->oo} -F(n)+(2+F(n+1))*(1+F(n+2))/(3+F(n+3)), where F = A000045 (Fibonacci numbers).

Apart from leading digits the same as A187798 and A187426. - R. J. Mathar, Sep 21 2013

			(13-5*sqrt(5))/2 = 0.9098300562505257589770658281718094113985...

Clark Kimberling, Table of n, a(n) for n = 0..999

Cf. A000045, A187426, A187798, A225667.

f[n_] := Fibonacci[n]; h[n_] := (2 + f[n + 1]) (1 + f[n + 2])/(3 + f[n + 3]); N[h[100] - f[100], 20]; d = RealDigits[(13 - 5*Sqrt[5])/2, 10, 120][[1]]

A229760 Decimal expansion of 25 - 10*sqrt(5).

Original entry on oeis.org

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

Offset: 1

Joost Gielen, Sep 28 2013

Apart from the first digit the same as A187799.

			2.639320225002103035908263312687237645593816403884742757291027545894790...

Ivan Panchenko, Table of n, a(n) for n = 1..1000

Cf. A187426, A187798, A094874.

RealDigits[25 - Sqrt@ 500, 10, 111][[1]] (* Robert G. Wilson v, Oct 01 2013 *)

25-sqrt(500) \\ Charles R Greathouse IV, Oct 01 2013

A081011 a(n) = Fibonacci(4n+3) + 2, or Fibonacci(2n+3)*Lucas(2n).

Original entry on oeis.org

4, 15, 91, 612, 4183, 28659, 196420, 1346271, 9227467, 63245988, 433494439, 2971215075, 20365011076, 139583862447, 956722026043, 6557470319844, 44945570212855, 308061521170131, 2111485077978052, 14472334024676223

Offset: 0

R. K. Guy, Mar 01 2003

For n>0, a(n) is the area of the trapezoid defined by the four points (F(n+1), F(n+2)), (F(n+2), F(n+1)), (F(n+3), F(n+4)), and (F(n+4), F(n+3)) where F(n) = A000045(n). - J. M. Bergot, May 14 2014

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

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Cf. A000045 (Fibonacci numbers), A000032 (Lucas numbers), A187798.

List([0..30], n-> Fibonacci(4*n+3) -2); # G. C. Greubel, Jul 14 2019

[Fibonacci(4*n+3)+2: n in [0..30]]; // Vincenzo Librandi, Apr 18 2011

with(combinat) for n from 0 to 30 do printf(`%d,`,fibonacci(4*n+3)+2) od # James Sellers, Mar 03 2003

Table[Fibonacci[4n+3] +2, {n,0,30}] (* or *)
Table[Fibonacci[2n+3]*LucasL[2n], {n, 0, 30}] (* Alonso del Arte, Apr 18 2011 *)
LinearRecurrence[{8,-8,1},{4,15,91},30] (* Harvey P. Dale, Apr 22 2017 *)

vector(30, n, n--; fibonacci(4*n+3)+2) \\ G. C. Greubel, Jul 14 2019

[fibonacci(4*n+3)+2 for n in (0..30)] # G. C. Greubel, Jul 14 2019

a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3).
G.f.: (4-17*x+3*x^2)/((1-x)*(1-7*x+x^2)). - Colin Barker, Jun 22 2012
Product_{n>=0} (1 - 1/a(n)) = (3-phi)/2 = A187798. - Amiram Eldar, Nov 28 2024

More terms from James Sellers, Mar 03 2003

A229780 Decimal expansion of (3+sqrt(5))/10.

Original entry on oeis.org

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

Offset: 0

Joost Gielen, Sep 29 2013

sqrt((3+sqrt(5))/10) = sqrt(phi^2/5) = (5+sqrt(5))/10 = (3+sqrt(5))/10 + 2/10 = 0.723606797... .
Essentially the same as A134972, A134945, A098317 and A002163. - R. J. Mathar, Sep 30 2013
Equals one tenth of the limit of (G(n+2)+G(n+1)+G(n-1)+G(n-2))/G(n), where G(n) is any nonzero sequence satisfying the recurrence G(n+1) = G(n) + G(n-1) including A000032 and A000045, as n --> infinity. - Richard R. Forberg, Nov 17 2014
3+sqrt(5) is the perimeter of a golden rectangle with a unit width. - Amiram Eldar, May 18 2021
Constant x such that x = sqrt(x) - 1/5. - Andrea Pinos, Jan 15 2024

			0.5236067977499...

Cf. A094874, A187798.

RealDigits[GoldenRatio^2/5,10,120][[1]] (* Harvey P. Dale, Dec 02 2014 *)

(3+sqrt(5))/10 = (phi/sqrt(5))^2 = phi^2/5 where phi is the golden ratio.

cp's OEIS Frontend

A019827 Decimal expansion of sin(Pi/10) (angle of 18 degrees).

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A019845 Decimal expansion of sine of 36 degrees.

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A094874 Decimal expansion of (5-sqrt(5))/2.

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

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A187426 Decimal expansion of (3-phi)^2 = 10 - 5*phi where phi is the golden ratio.

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A187799 Decimal expansion of 20/phi^2, where phi is the golden ratio. Also (with a different offset), decimal expansion of 3 - sqrt(5).

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