A020837 -id:A020837 - cp's OEIS frontend

A296184 Decimal expansion of 2 + phi, with the golden section phi from A001622.

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

Offset: 1

Wolfdieter Lang, Jan 08 2018

In a regular pentagon, inscribed in a unit circle this equals twice the largest distance between a vertex and a midpoint of a side.
This is an integer in the quadratic number field Q(sqrt(5)).
Only the first digit differs from A001622.

			3.618033988749894848204586834365638117720309179805762862135448622705260462...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.25, p. 417.

Sumit Kumar Jha, Two complementary relations for the Rogers-Ramanujan continued fraction, arXiv:2112.12081 [math.NT], 2021.
Index entries for algebraic numbers, degree 2.

Cf. A001622, A094214, A104457, A176055, A020837.

2 + 2*cos(2*Pi/n): A104457 (n = 5), A116425 (n = 7), A332438 (n = 9), A019973 (n = 12).

First@ RealDigits[2 + GoldenRatio, 10, 77] (* Michael De Vlieger, Jan 13 2018 *)

(5 + sqrt(5))/2 \\ Altug Alkan, Mar 19 2018

Equals 2 + A001622 = 1 + A104457 = 3 + A094214.
From Christian Katzmann, Mar 19 2018: (Start)
Equals Sum_{n>=0} (15*(2*n)!+40*n!^2)/(2*n!^2*3^(2*n+2)).
Equals 5/2 + Sum_{n>=0} 5*(2*n)!/(2*n!^2*3^(2*n+1)). (End)
Constant c = 2 + 2*cos(2*Pi/10). The linear fractional transformation z -> c - c/z has order 10, that is, z = c - c/(c - c/(c - c/(c - c/(c - c/(c - c/(c - c/(c - c/(c - c/(c - c/(z)))))))))). - Peter Bala, May 09 2024

A079585 Decimal expansion of c = (7-sqrt(5))/2.

Original entry on oeis.org

2, 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

Benoit Cloitre, Jan 26 2003

c is an integer in the quadratic number field Q(sqrt(5)). - Wolfdieter Lang, Jan 08 2018
From Amiram Eldar, Jul 16 2021: (Start)
Sum_{k>=0} 1/F(2^k) is sometimes called "Millin series" after D. A. Millin, a high school student at Annville, Pennsylvania, who posed in 1974 the problem of proving that it equals (7-sqrt(5))/2. This identity was in fact already known to Lucas in 1878.
Mahler (1975) provided a false proof that this sum is transcendental. The mistake was corrected in Mahler (1976). (End)
The name "Millin" was a misprint of "Miller", the author of the problem was Dale A. Miller. His name was corrected in the solution to the problem (1976). - Amiram Eldar, Feb 29 2024

			c = 2.3819660112501051517954131656343618822796908201942371378645513772947...

Jean-Paul Allouche and Jeffrey Shallit, Automatic sequences, Cambridge University Press, 2003, p. 65.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Section 1.2, p. 7.
Ross Honsberger, Mathematical Gems III, Washington, DC: Math. Assoc. Amer., 1985, pp. 135-137.
Alfred S. Posamentier and Ingmar Lehmann, [Phi], The Glorious Golden Ratio, Prometheus Books, 2011, page 75.

Chai Wah Wu, Table of n, a(n) for n = 1..10001
I. J. Good, A Reciprocal Series of Fibonacci Numbers, Fib. Quart., Vol. 12, No. 4 (1974), p. 346.
History of Science and Mathematics StackExchange, Who was D.A. Millin, the eponym of the Millin Series?, 2022.
Edouard Lucas, Théorie des Fonctions Numériques Simplement Périodiques. [Continued], American Journal of Mathematics, Vol. 1, No. 3 (1878), pp. 197-240. See p. 225, equations 125 and 127.
Kurt Mahler, On the transcendency of the solutions of special class of functional equations, Bull. Austral. Math. Soc., Vol. 13, No. 3 (1975), pp. 389-410.
Kurt Mahler, On the transcendency of the solutions of a special class of functional equations: Corrigendum, Bull. Austral. Math. Soc., Vol. 14, No. 3 (1976), pp. 477-478.
Dale Miller, Publications.
D. A. Millin, Problem H-237, The Fibonacci Quarterly, Vol. 12, No. 3 (1974), p. 309; Sum Reciprocal!, Solution to Problem H-237 by A. G. Shannon, ibid., Vol. 14, No. 2 (1976), pp. 186-187.
Michael Penn, The Millin Series (A nice Fibonacci sum), YouTube video, 2020.
Proofwiki, Definition:Millin Series.
Stanley Rabinowitz, A note on the sum 1/w_{k2^n}, Missouri J. Math. Sci., Vol. 10, No. 3 (1998), pp. 141-146.
Eric Weisstein's World of Mathematics, Millin Series.
Index entries for algebraic numbers, degree 2

Cf. A001622, A020837, A058635.

RealDigits[4 - GoldenRatio, 10, 111][[1]] (* Robert G. Wilson v, Jan 31 2012 *)

(7 - sqrt(5))/2 \\ Michel Marcus, Sep 05 2017

c = (7-sqrt(5))/2 = 4 - phi, with phi from A001622.
c = 7/2 - 10*A020837.
c = Sum_{k>=0} 1/F(2^k), where F(k) denotes the k-th Fibonacci number; c = Sum_{k>=0} 1/A058635(k).
Periodic continued fraction representation is [2, 2, 1, 1, 1, 1, ....]. - R. J. Mathar, Mar 24 2011
Minimal polynomial: 11 - 7*x + x^2. - Stefano Spezia, Oct 16 2024

A176055 Decimal expansion of (2+sqrt(5))/2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 07 2010

Continued fraction expansion of (2+sqrt(5))/2 is A010698.

a(n) = A020837(n-1) for n > 1; a(1) = 2.

			2.11803398874989484820...

Index entries for algebraic numbers, degree 2.

Cf. A002163 (sqrt(5)), A020837 (1/sqrt(80)), A010698 (repeat 2, 8).

Cf. A000045, A001622.

RealDigits[GoldenRatio + 1/2, 10, 100][[1]] (* Amiram Eldar, Jun 06 2021 *)

Equals 1/2 + phi, with phi = A001622.
From Amiram Eldar, Jun 06 2021: (Start)
Equals 1 + Sum{k>=0} 1/(Fibonacci(2*k+1)+1).
Equals 1 + Sum{k>=0} binomial(2*k,k)/20^k. (End)

A204188 Decimal expansion of sqrt(5)/4.

Original entry on oeis.org

5, 5, 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, 7

Offset: 0

Jonathan Sondow, Jan 14 2012

Equals Product_{n>=1} (1 - 1/A000032(2^n)).
Essentially the same as A019863 and A019827. - R. J. Mathar, Jan 16 2012
The value is the distance of the W point of the Wigner-Seitz cell of the body-centered cubic lattice (that is the Brioullin zone of the face-centered cubic lattice) to its four nearest neighbors. Let the points of the simple cubic lattice be at (1,0,0), (0,1,0), (1,0,0) etc and the point in the cube center at (1/2, 1/2, 1/2). Then W is at (0, 1/4, 1/2) [or any of the 24 symmetry related positions like (0, 3/4, 1/2), (0, 1/2, 1/4) etc.], and the four lattice points closest to W are at (-1/2, 1/2, 1/2), (0,0,0), (1/2, 1/2, 1/2) and (0,0,1). - R. J. Mathar, Aug 19 2013

			0.5590169943749474241022934171828190588601545899028814310677243113526302...

J. Sondow, Evaluation of Tachiya's algebraic infinite products involving Fibonacci and Lucas numbers, Diophantine Analysis and Related Fields 2011 - AIP Conference Proceedings, vol. 1385, pp. 97-100.
Y. Tachiya, Transcendence of certain infinite products, J. Number Theory 125 (2007), 182-200.
Wikipedia, Brillouin zone

Cf. A001622, A002163.

Maple
```
evalf(sqrt(5)/4);
```

RealDigits[Sqrt[5]/4, 10, 100][[1]] (* Amiram Eldar, Dec 04 2018 *)

sqrt(5)/4 \\ Charles R Greathouse IV, Apr 21 2016

Equals sqrt(5)/4 = (-1 + 2*phi)/4, with the golden section phi from A001622.

Equals 5*A020837.

A208899 Decimal expansion of sqrt(5)/3.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Mar 03 2012

Equals the absolute value of the cosine of the dihedral angle between two adjacent faces of the regular icosahedron.

			0.7453559924...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Mathematics Stack Exchange, Maximum and Minimum Value of f(x)
Michael Penn, an "extreme" extreme value problem., YouTube video, 2022.
Wikipedia, Exsphere

Cf. A002163, A020837.

RealDigits[Sqrt[5]/3,10,120][[1]] (* Harvey P. Dale, Jul 03 2013 *)

first(n) = default(realprecision, max(28, n+10)); digits(((sqrt(5)/3)*10^n)\1) \\ David A. Corneth, Dec 19 2022

Equals A002163/3 = 20*A020837/3.

Equals Sum_{k>=0} (-1)^k * binomial(2*k,k)/5^k. - Amiram Eldar, Aug 03 2020

A242703 Decimal expansion of sqrt(7)/2.

Original entry on oeis.org

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

Offset: 1

Alonso del Arte, May 20 2014

Absolute value of the imaginary part of any of the nontrivial divisors of 2 in O_Q(sqrt(-7)).
The incircle of a triangle with sides of lengths 4, 5, 6 units respectively has a radius of sqrt(7)/2.
With a different offset, decimal expansion of 5 * sqrt(7).
From Wolfdieter Lang, Nov 18 2017: (Start)
In a regular hexagon inscribed in a circle with a radius of 1 unit the three distinct distances between any vertex and the middle of the sides are 1/2, sqrt(7)/2 and sqrt(13)/2.
The continued fraction expansion of sqrt(7)/2 is 1, repeat(3, 10, 3, 2). The convergents are given in A294972/A294973. (End)

			1.32287565553229529525...

Iain Fox, Table of n, a(n) for n = 1..20000 (first 1000 terms from Ivan Panchenko)

Cf. A010527, A020837, A040022, A294972/A294973.

Mathematica
```
RealDigits[Sqrt[7]/2, 10, 100][[1]]
```

{ default(realprecision, 20080); x=sqrt(7)/2; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b242703.txt", n, " ", d)); } \\ Iain Fox, Nov 18 2017

(1/2 - sqrt(-7)/2)(1/2 + sqrt(-7)/2) = 2.

Equals A010465/2. - R. J. Mathar, Feb 23 2021

A119032 a(n+2) = 18*a(n+1) - a(n) + 8.

Original entry on oeis.org

0, 9, 170, 3059, 54900, 985149, 17677790, 317215079, 5692193640, 102142270449, 1832868674450, 32889493869659, 590178020979420, 10590314883759909, 190035489886698950, 3410048503076821199, 61190837565496082640, 1098025027675852666329, 19703259660599851911290

Offset: 1

Richard Choulet, Aug 30 2007, Oct 09 2007

Arises in calculating A107075. A053606 follows the same recurrence.

Index entries for linear recurrences with constant coefficients, signature (19,-19,1).

Cf. A000032, A020837, A053606, A107075.

LinearRecurrence[{19, -19, 1}, {0, 9, 170}, 20] (* Amiram Eldar, Dec 02 2024 *)

a(n+1) = 9*a(n+1) + 4 + (80*a(n)^2+80*a(n)+25)^(1/2).
G.f.: (9*x-x^2)/((1-x)*(1-18*x+x^2)).
a(n) = ((sqrt(5)+2)/8)*(9+4*sqrt(5))^(n-1) + ((-sqrt(5)+2)/8)*(9-4*sqrt(5))^(n-1) - 1/2. - Richard Choulet, Nov 26 2008
a(n) = (Lucas(6*n-3)-4)/8, where Lucas(n) = A000032(n). - Gary Detlefs, Dec 07 2010
Product_{n>=2} (1 + 1/a(n)) = sqrt(5)/2 (= 10 * A020837). - Amiram Eldar, Dec 02 2024

A041143 Denominators of continued fraction convergents to sqrt(80).

Original entry on oeis.org

1, 1, 17, 18, 305, 323, 5473, 5796, 98209, 104005, 1762289, 1866294, 31622993, 33489287, 567451585, 600940872, 10182505537, 10783446409, 182717648081, 193501094490, 3278735159921, 3472236254411, 58834515230497, 62306751484908, 1055742538989025

Offset: 0

N. J. A. Sloane

This is the sequence of Lehmer numbers U_n(sqrt(R),Q) for the parameters R = 16 and Q = -1; it is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for all positive integers n and m. Consequently, this is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, May 28 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..199 [First term 0 removed by _Georg Fischer_, Jul 01 2019]
Eric W. Weisstein, MathWorld: Lehmer Number
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (0,18,0,-1).

Cf. A041142, A020837, A040071, A010532, A001076, A002530.

List([0..30], n-> (5 +3*(-1)^n)*Fibonacci(3*(n+1))/16 ); # G. C. Greubel, Jul 02 2019

I:=[1,1,17,18]; [n le 4 select I[n] else 18*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 11 2013

with(numtheory): cf := cfrac(sqrt(80),25): seq(nthdenom(cf,n), n=0..24); # Peter Luschny, Jul 06 2019

Denominator/@Convergents[Sqrt[80], 30] (* Vladimir Joseph Stephan Orlovsky, Jul 05 2011 *)
CoefficientList[Series[(1 + x - x^2)/(1 - 18 x^2 + x^4), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 11 2013 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,0,18,0]^n*[1;1;17;18])[1,1] \\ Charles R Greathouse IV, Nov 13 2015

a(n) = (5 + 3*(-1)^n)*fibonacci(3*(n+1))/16 \\ Georg Fischer, Jul 01 2019

from sympy import continued_fraction_convergents as convergents, continued_fraction_iterator as cf, sqrt, denom
denominators = (denom(c) for c in convergents(cf(sqrt(80))))
print([next(denominators) for  in range(30)]) # _Ehren Metcalfe, Jul 03 2019

[(5 +3*(-1)^n)*fibonacci(3*(n+1))/16 for n in (0..30)] # G. C. Greubel, Jul 02 2019

G.f.: (1 + x - x^2) / (1 - 18*x^2 + x^4).
a(n) = 18*a(n-2) - a(n-4).
From Peter Bala, May 28 2014: (Start)
Let alpha = 2 + sqrt(5) and beta = 2 - sqrt(5) be the roots of the equation x^2 - 4*x - 1 = 0. Then a(n) = (alpha^n - beta^n)/(alpha - beta) for n even, while a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2) for n odd.
a(n) = A001076(n+1) for n even; a(n) = 1/4*A001076(n+1) for n odd.
a(n) = Product_{k = 1..floor(n/2)} ( 16 + 4*cos^2(k*Pi/(n+1)) ).
Recurrence equations: a(0) = 1, a(1) = 1 and for n >= 1, a(2*n) = 16*a(2*n - 1) + a(2*n - 2) and a(2*n + 1) = a(2*n) + a(2*n - 1). (End)
a(n) = (5 + 3*(-1)^n)*Fibonacci(3*(n+1))/16. - Ehren Metcalfe, Apr 15 2019

First term 0 removed from b-file, formulas and programs by Georg Fischer, Jul 01 2019

A138373 Count of post-period decimal digits up to which the rounded n-th convergent to sqrt(5)/2 agrees with the exact value.

Original entry on oeis.org

1, 3, 3, 5, 6, 8, 9, 10, 9, 13, 13, 15, 15, 18, 19, 20, 22, 22, 24, 25, 26, 27, 29, 30, 32, 33, 34, 35, 36, 38, 39, 40, 41, 43, 44, 45, 47, 47, 49, 50, 52, 53, 54, 55, 56, 58, 57, 60, 61, 62, 64, 64, 67, 68, 68, 71, 72, 73, 74, 75, 76, 78, 78, 80, 82, 83, 84, 85, 86, 88, 88, 90

Offset: 1

Artur Jasinski, Mar 17 2008

Defined equivalent to A138367 considering the constant 1.1188.. = 10*A020837 and its convergents 9/8 (n=1), 19/17 (n=2), 161/144 (n=3), 341/305 (n=4), 2889/2584 (n=5).

Cf. A138335 - A138339, A138343, A138366, A138367, A138369, A138370 - A138374.

Definition and values replaced as defined via continued fractions - R. J. Mathar, Oct 01 2009

A352484 Decimal expansion of the probability that when three real numbers are chosen at random, uniformly and independently in the interval [0,1], they can be the lengths of the sides of a triangle whose altitudes are also the sides of some triangle.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Mar 18 2022

Without the condition on the altitudes the probability is 1/2.

			0.30583672225078887563435958170197817216032242014342...

Mohammed Yaseen, Table of n, a(n) for n = 0..10000
Murray S. Klamkin, Problem 1494, Crux Mathematicorum, Vol. 15, No. 10 (1989), p. 298; Solution to Problem 1494, by P. Penning, ibid., Vol. 17, No. 2 (1991), pp. 53-54.
Eric Weisstein's World of Mathematics, Altitude.
Index entries for transcendental numbers.

Cf. A020837, A134972, A352485.

RealDigits[2*Log[Sqrt[5] - 1] + 1 - Sqrt[5]/2, 10, 100][[1]]

2*log(sqrt(5)-1) + 1 - sqrt(5)/2 \\ Charles R Greathouse IV, Nov 26 2024

Equals 2*log(sqrt(5)-1) + 1 - sqrt(5)/2.

cp's OEIS Frontend

A296184 Decimal expansion of 2 + phi, with the golden section phi from A001622.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A079585 Decimal expansion of c = (7-sqrt(5))/2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A176055 Decimal expansion of (2+sqrt(5))/2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A204188 Decimal expansion of sqrt(5)/4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A208899 Decimal expansion of sqrt(5)/3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A242703 Decimal expansion of sqrt(7)/2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A119032 a(n+2) = 18*a(n+1) - a(n) + 8.

Views

Author

Keywords

Comments