A296184 -id:A296184 - cp's OEIS frontend

A344476 Decimal expansion of the value of the Buchstab function at phi + 2 = (5 + sqrt(5))/2 (A296184).

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

Offset: 0

Amiram Eldar, May 20 2021

			0.56093863969277163346004116368055699296113179704964...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 286.

A. A. Bukhstab, Asymptotic estimates of a general number-theoretic function (in Russian), Matematicheskii Sbornik, Vol. 44, No. 6 (1937) pp. 1239-1246.
Pieter Moree, A special value of Dickman's function, Math. Student, Vol. 64 (1995), pp. 47-50; entire volume.
Eric Weisstein's World of Mathematics, Buchstab Function.
Wikipedia, Buchstab function.

Cf. A000796, A001622, A002390, A013661, A296184, A344475.

RealDigits[(1 + 2*Log[GoldenRatio] + Log[GoldenRatio]^2 - Pi^2/60)/(GoldenRatio + 2), 10, 100][[1]]

my(phi = quadgen(5)); (1 + 2*log(phi) + log(phi)^2 - Pi^2/60)/(phi+2) \\ Amiram Eldar, Jan 09 2025

Equals (1 + 2*log(phi) + log(phi)^2 - Pi^2/60)/(phi+2) (Moree, 1995).

More terms from Amiram Eldar, Jan 09 2025

A104457 Decimal expansion of 1 + phi = phi^2 = (3 + sqrt(5))/2.

Original entry on oeis.org

2, 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, 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

Offset: 1

Eric W. Weisstein, Mar 08 2005

Only first term differs from the decimal expansion of phi.
Zelo extends work of D. Roy by showing that the square of the golden ratio is the optimal exponent of approximation by algebraic numbers of degree 4 with bounded denominator and trace. - Jonathan Vos Post, Mar 02 2009 (Cf. last sentence in the Zelo reference. - Joerg Arndt, Jan 04 2014)
Hawkes asks: "What two numbers are those whose product, difference of their squares, and the ratio or quotient of their cubes, are all equal to each other?". - Charles R Greathouse IV, Dec 11 2012
This is the case n=10 in (Gamma(1/n)/Gamma(3/n))*(Gamma((n-1)/n)/Gamma((n-3)/n)) = 1+2*cos(2*Pi/n). - Bruno Berselli, Dec 14 2012
An algebraic integer of degree 2, with minimal polynomial x^2 - 3x + 1. - Charles R Greathouse IV, Nov 12 2014 [The other root is 2 - phi = A132338 - Wolfdieter Lang, Aug 29 2022]
To eight digits: 5*(((Pi+1)/e)-1) = 2.61803395481182... - Dan Graham, Nov 21 2017
The ratio diagonal/side of the second smallest diagonal in a regular 10-gon. - Mohammed Yaseen, Nov 04 2020
phi^2/10 is the moment of inertia of a solid regular icosahedron with a unit mass and a unit edge length (see A341906). - Amiram Eldar, Jun 08 2021

			2.6180339887498948482045868343656381177203091798...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.17.1, p. 153.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 138-139.
Damien Roy. Diophantine Approximation in Small Degree. Centre de Recherches Mathématiques. CRM Proceedings and Lecture Notes. Volume 36 (2004), 269-285.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 45.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Murray Berg, Phi, the golden ratio (to 4599 decimal places) and Fibonacci numbers, Fibonacci Quarterly, Vol. 4, No. 2 (1966), pp. 157-162.
John Hawkes et al., Question 1029, The Mathematical Questions Proposed in the Ladies' Diary (1817), p. 339. Originally published 1798 and answered in 1799.
Casey Mongoven, Phi^2 number 1; electronic music created using Phi^2.
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.
Damien Roy, Diophantine Approximation in Small Degree, arXiv:math/0303150 [math.NT], 2003.
Eric Weisstein's World of Mathematics, Chromatic Polynomial.
Eric Weisstein's World of Mathematics, Fibonacci Hyperbolic Functions.
Wikipedia, Perron number.
Dmitrij Zelo, Simultaneous Approximation to Real and p-adic Numbers, arXiv:0903.0086 [math.NT], 2009.
Index entries for algebraic numbers, degree 2.
Index entries for sequences related to moment of inertia.

Cf. A001622, A094214, A094885, A132338, A229780, A239798, A341906.
Cf. A001519, A001906, A032908, A049310.
2 + 2*cos(2*Pi/n): A116425 (n = 7), A332438 (n = 9), A296184 (n = 10), A019973 (n = 12).

SetDefaultRealField(RealField(100)); (1+Sqrt(5))^2/4; // G. C. Greubel, Nov 23 2018

RealDigits[N[GoldenRatio+1,200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2011 *)

(3+sqrt(5))/2 \\ Charles R Greathouse IV, Aug 21 2012

numerical_approx(golden_ratio^2, digits=100) # G. C. Greubel, Nov 23 2018

Equals 2 + A094214 = 1 + A001622. - R. J. Mathar, May 19 2008
Satisfies these three equations: x-sqrt(x)-1 = 0; x-1/sqrt(x)-2 = 0; x^2-3*x+1 = 0. - Richard R. Forberg, Oct 11 2014
Equals the nested radical sqrt(phi^2+sqrt(phi^4+sqrt(phi^8+...))). For a proof, see A094885. - Stanislav Sykora, May 24 2016
From Christian Katzmann, Mar 19 2018: (Start)
Equals Sum_{n>=0} (5*(2*n)!+8*n!^2)/(2*n!^2*3^(2*n+1)).
Equals 3/2 + Sum_{n>=0} 5*(2*n)!/(2*n!^2*3^(2*n+1)). (End)
Equals 1/A132338 = 2*A239798 = 5*A229780. - Mohammed Yaseen, Nov 04 2020
Equals Product_{k>=1} 1 + 1/(phi + phi^k), where phi is the golden ratio (A001622) (Ohtsuka, 2018). - Amiram Eldar, Dec 02 2021
c^n = phi * A001906(n) + A001519(n), where c = phi^2. - Gary W. Adamson, Sep 08 2023
Equals lim_{n->oo} S(n, 3)/S(n-1, 3) with the S-Chebyshev polynomials (see A049310), S(3, n) = A000045(2*(n+1)) = A001906(n+1). - Wolfdieter Lang, Nov 15 2023
From Peter Bala, May 08 2024: (Start)
Constant c = 2 + 2*cos(2*Pi/5).
The linear fractional transformation z -> c - c/z has order 5, that is, z = c - c/(c - c/(c - c/(c - c/(c - c/z)))). (End)
Equals Product_{k>=1} (1 + 1/A032908(k)). - Amiram Eldar, Nov 28 2024

A030191 Scaled Chebyshev U-polynomial evaluated at sqrt(5)/2.

Original entry on oeis.org

1, 5, 20, 75, 275, 1000, 3625, 13125, 47500, 171875, 621875, 2250000, 8140625, 29453125, 106562500, 385546875, 1394921875, 5046875000, 18259765625, 66064453125, 239023437500, 864794921875, 3128857421875, 11320312500000, 40957275390625, 148184814453125

Offset: 0

Wolfdieter Lang

Number of (s(0), s(1), ..., s(2n+4)) such that 0 < s(i) < 10 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+4, s(0) = 1, s(2n+4) = 5. - Herbert Kociemba, Jun 14 2004
Binomial transform of A002878. - Philippe Deléham, Oct 04 2005
Diagonal of square array A216219. - Philippe Deléham, Mar 15 2013
Lim_{n->oo} a(n+1)/a(n) = 2 + phi = A296184, where phi = A001622. - Wolfdieter Lang, Nov 16 2023~

			G.f. = 1 + 5*x + 20*x^2 + 75*x^3 + 275*x^4 + 1000*x^5 + 3625*x^6 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Santiago Alzate, Oscar Correa, and Rigoberto Flórez, Fibonacci identities from Jordan Identities, arXiv:2009.02639 [math.NT], 2020.
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0 ,b=1; p=5, q=-5.
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eqs. (38) and (45), lhs, m=5.
László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
Index entries for linear recurrences with constant coefficients, signature (5,-5).
Index entries for sequences related to Chebyshev polynomials.

Cf. A000032, A000045, A001622.

Cf. A002878, A039717, A081567, A109466, A216219.

a:=[1,5];; for n in [3..30] do a[n]:=5*(a[n-1]-a[n-2]); od; a; # G. C. Greubel, Dec 28 2019

I:=[1,5]; [n le 2 select I[n] else 5*(Self(n-1) - Self(n-2)): n in [1..30]]; // G. C. Greubel, Dec 28 2019

seq(coeff(series(1/(1-5*x+5*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Dec 28 2019

Table[MatrixPower[{{2,1},{1,3}},n][[1]][[2]],{n,0,44}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
a[ n_]:= (((5 + Sqrt[5])/2)^(n + 1) - ((5 - Sqrt[5])/2)^(n + 1)) / Sqrt[5] // Expand; (* Michael Somos, Aug 27 2015 *)
Table[If[EvenQ[n], 5^(n/2)*LucasL[n+1], 5^((n+1)/2)*Fibonacci[n+1]], {n,0,30}] (* G. C. Greubel, Dec 28 2019 *)

{a(n) = imag((quadgen(5) + 2)^(n+1))}; /* Michael Somos, Aug 27 2015 */

[lucas_number1(n,5,5) for n in range(1, 22)] # Zerinvary Lajos, Apr 22 2009

a(n) = (sqrt(5))^n*U(n, sqrt(5)/2).
G.f.: 1/(1 - 5*x + 5*x^2).
a(2*k+1) = 5^(k+1)*Fibonacci(2*k+2).
a(2*k) = 5^k*Lucas(2*k+1).
a(n-1) = Sum_{k=0..n} C(n, k)*Fibonacci(2*k). - Benoit Cloitre, Jun 21 2003
a(n) = 5*a(n-1) - 5*a(n-2). - Benoit Cloitre, Oct 23 2003
a(n-1) = (((5+sqrt(5))/2)^n - ((5-sqrt(5))/2)^n)/sqrt(5) is the 2nd binomial transform of Fibonacci(n), the first binomial transform of Fibonacci(2n) and its n-th term is the n-th term of the third binomial transform of Fibonacci(3n) divided by 2^n. - Paul Barry, Mar 23 2004
a(n) = Sum_{k-0..n} 5^k*A109466(n,k). - Philippe Deléham, Nov 28 2006
a(n) = 5*A039717(n), n>0. - Philippe Deléham, Mar 12 2013
a(n) = A216219(n,n+3) = A216219(n,n+4) = A216219(n+3,n) = A216219(n+4,n). - Philippe Deléham, Mar 15 2013
G.f.: 1/(1-5*x/(1+x/(1-x))). - Philippe Deléham, Mar 15 2013
a(n) = -a(-2-n) * 5^(n+1) for all n in Z. - Michael Somos, Aug 27 2015
E.g.f.: exp((5-sqrt(5))*x/2)*((5 + sqrt(5))*exp(sqrt(5)*x) - 5 + sqrt(5))/(2*sqrt(5)). - Stefano Spezia, Dec 29 2019
a(n) = Sum_{k=0..n} A081567(n-k)*2^k. - Philippe Deléham, Mar 10 2023

A116425 Decimal expansion of 2 + 2cos(2Pi/7).

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Feb 15 2006

A root of the equation x^3 - 5*x^2 + 6*x - 1 = 0. - Arkadiusz Wesolowski, Jan 13 2016

The other two roots of this minimal polynomial of the present algebraic number (rho(7))^2, with rho(7) = 2*cos(Pi/7) = A160389 are (2*cos(3*Pi/7))^2 = (A255241)^2 and (2*cos(5*Pi/7))^2 = (-A255249)^2. - Wolfdieter Lang, Mar 30 2020

			3.246979603717467061...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.25 Tutte-Beraha Constants, p. 417.

Jesús Salas and Alan D. Sokal, Transfer matrices and partition functions zeros for antiferromagnetic Potts models, arXiv:cond-mat/0004330 [cond-mat.stat-mech], 2000-2001, p. 64.
Eric Weisstein's World of Mathematics, Logistic Map
Eric Weisstein's World of Mathematics, Silver Constant
Index entries for algebraic numbers, degree 3

Cf. A231187, A160389.
Cf. A003558, A054142, A255241, A255249.
2 + 2*cos(2*Pi/n): A104457 (n = 5), A332438 (n = 9), A296184 (n = 10), A019973 (n = 12).

First@ RealDigits[N[2 + 2 Cos[2 Pi/7], 120]] (* Michael De Vlieger, Jan 13 2016 *)

2 + 2*cos(2*Pi/7) \\ Michel Marcus, Jan 13 2016

Equals (2*cos(Pi/7))^2 = (A160389)^2.
Equals 2 + i^(4/7) - i^(10/7). - Peter Luschny, Apr 04 2020
Let  c = 2 + 2*cos(2*Pi/7). The linear fractional transformation z -> c - c/z has order 7, that is, z = c - c/(c - c/(c - c/(c - c/(c - c/(c - c/(c - c/z)))))). - Peter Bala, May 09 2024

A332438 Decimal expansion of (2*cos(Pi/9))^2 = A332437^2.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Mar 31 2020

This algebraic number rho(9)^2 of degree 3 is a root of its minimal polynomial x^3 - 6*x^2 + 9*x - 1.
The other two roots are x2 = (2*cos(5*Pi/9))^2 = (2*cos(4*Pi/9))^2 = (R(4,rho(9)))^2 = 2 - rho(9) = 0.120614758..., and x3 = (2*cos(7*Pi/9))^2 = (2*cos(7*Pi/9))^2 = (R(7,rho(9)))^2 = 4 + rho(9) - rho(9)^2 = 2.347296355... = A130880 + 2, with rho(9) = 2*cos(Pi/9) = A332437, the monic Chebyshev polynomials R (see A127672), and the computation is done modulo the minimal polynomial of rho(9) which is x^3 - 3*x - 1 (see A187360).
This gives the representation of these roots in the power basis of the simple field extension Q(rho(9)). See the linked W. Lang paper in A187360, sect. 4.
This number rho(9)^2 appears as limit of the quotient of consecutive numbers af various sequences, e.g., A094256 and A094829.
The algebraic number rho(9)^2 - 2 = 1.532088898... of Q(rho(9)) has minimal polynomial x^3 - 3*x + 1 over Q. The other roots are -rho(9) = -A332437 and 2 + rho(9) - rho(9)^2 = A130880. - Wolfdieter Lang, Sep 20 2022

			3.5320888862379560704047853011108333478716649...

Index entries for algebraic numbers, degree 3.

Cf. A019879, A063289, A094256, A094829, A127672, A130880, A187360, A332437.

2 + 2*cos(2*Pi/n): A104457 (n = 5), A116425 (n = 7), A296184 (n = 10), A019973 (n = 12).

RealDigits[(2*Cos[Pi/9])^2, 10, 100][[1]] (* Amiram Eldar, Mar 31 2020 *)

(2*cos(Pi/9))^2 \\ Michel Marcus, Sep 23 2022

Equals (2*cos(Pi/9))^2 = rho(9)^2 = A332437^2.
Equals 2 + i^(4/9) - i^(14/9). - Peter Luschny, Apr 04 2020
Equals 2 + w1^(1/3) + w2^(1/3), where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) and w2 = (-1 - sqrt(3)*i)/2 are the complex roots of x^3 - 1. - Wolfdieter Lang, Sep 20 2022
Constant c = 2 + 2*cos(2*Pi/9). The linear fractional transformation z -> c - c/z has order 9, that is, z = c - c/(c - c/(c - c/(c - c/(c - c/(c - c/(c - c/(c - c/(c - c/(z))))))))). - Peter Bala, May 09 2024
From Amiram Eldar, Nov 22 2024: (Start)
Equals 3 + sec(Pi/9)/2 = 3 + 1/(2*A019879).
Equals 3 + Product_{k>=3} (1 + (-1)^k/A063289(k)). (End)

A351415 Intersection of Beatty sequences for (1+sqrt(5))/2 and sqrt(5).

Original entry on oeis.org

4, 6, 8, 11, 17, 22, 24, 29, 33, 35, 38, 40, 42, 46, 51, 53, 55, 58, 64, 67, 69, 71, 76, 80, 82, 84, 87, 93, 98, 100, 105, 111, 114, 116, 118, 122, 127, 129, 131, 134, 140, 145, 147, 152, 156, 158, 160, 163, 165, 169, 174, 176, 181, 187, 190, 192, 194, 199

Offset: 1

Clark Kimberling, Feb 10 2022

nonn

Conjecture: every term of the difference sequence is in {2,3,4,5,6}, and each occurs infinitely many times.
From Clark Kimberling, Jul 29 2022: (Start)
This is the first of four sequences that partition the positive integers. Starting with a general overview, suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers.  Let u' and v' be their complements, and assume that the following four sequences are infinite:
(1)  u ^ v = intersection of u and v (in increasing order);
(2)  u ^ v';
(3)  u' ^ v;
(4)  u' ^ v'.
Every positive integer is in exactly one of the four sequences. For A351415, u, v, u', v', are the Beatty sequences given by u(n) = floor(n*(1+sqrt(5))/2) and v(n) = floor(n*sqrt(5)), so that r = (1+sqrt(5))/2, s = sqrt(5), r' = (3+sqrt(5))/2, s' = (5 + sqrt(5))/4.
(1)  u ^ v = (4, 6, 8, 11, 17, 22, 24, 29, 33, 35, 38, 40, 42, ...) = A351415
(2)  u ^ v' = (1, 3, 9, 12, 14, 16, 19, 21, 25, 27, 30, 32, ...) =    A356101
(3)  u' ^ v = (2, 13, 15, 20, 26, 31, 44, 49, 60, 62, 73, 78, ...) =  A356102
(4)  u' ^ v' = (5, 7, 10, 18, 23, 28, 34, 36, 39, 41, 47, 52, ...) =  A356103
(End)

			The two Beatty sequences are (1,3,4,6,8,9,11,12,14,...) and (2,4,6,8,11,13,15,17,...), with common terms forming the sequence (4,6,8,11,...).

Cf. A000201, A022839, A296184, A346308, A347469, A347793.

Cf. A001950, A108598, A356101, A356102, A356103, A356104 (results of composition instead of intersections), A190509 (composites, reversed order).

z = 200;
r = (1 + Sqrt[5])/2; u = Table[Floor[n*r], {n, 1, z}]  (* A000201 *)
u1 = Take[Complement[Range[1000], u], z]  (* A001950 *)
r1 = Sqrt[5]; v = Table[Floor[n*r1], {n, 1, z}]  (* A022839 *)
v1 = Take[Complement[Range[1000], v], z]  (* A108598 *)
Intersection[u, v]    (* A351415 *)
Intersection[u, v1]   (* A356101 *)
Intersection[u1, v]   (* A356102 *)
Intersection[u1, v1]  (* A356103 *)

A188593 Decimal expansion of (diagonal)/(shortest side) of a golden rectangle.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 04 2011

A rectangle of length L and width W is a golden rectangle if L/W = r = (1+sqrt(5))/2. The diagonal has length D = sqrt(L^2+W^2), so D/W = sqrt(r^2+1) = sqrt(r+2).
Largest root of x^4 - 5x^2 + 5. - Charles R Greathouse IV, May 07 2011
This is the case n=10 of (Gamma(1/n)/Gamma(2/n))*(Gamma((n-1)/n)/Gamma((n-2)/n)) = 2*cos(Pi/n). - Bruno Berselli, Dec 13 2012
Edge length of a pentagram (regular star pentagon) with unit circumradius. - Stanislav Sykora, May 07 2014
The ratio diagonal/side of the shortest diagonal in a regular 10-gon. - Mohammed Yaseen, Nov 04 2020

			1.902113032590307144232878666758764286811397268251...

Chai Wah Wu, Table of n, a(n) for n = 1..10001
Michael Penn, On the fifth root of the identity matrix, YouTube video, 2022.
Eric Weisstein's World of Mathematics, Golden Rectangle.
Eric Weisstein's World of Mathematics, Pentagram.
Index entries for algebraic numbers, degree 4.

Cf. A001622 (decimal expansion of the golden ratio), A019881.
Cf. A188594 (D/W for the silver rectangle, r=1+sqrt(2)).
Cf. A090771, A195693, A195723, A296184.

SetDefaultRealField(RealField(100)); Sqrt((5+Sqrt(5))/2); // G. C. Greubel, Nov 02 2018

r = (1 + 5^(1/2))/2; RealDigits[(2 + r)^(1/2), 10, 130][[1]]
RealDigits[Sqrt[GoldenRatio+2],10,130][[1]] (* Harvey P. Dale, Oct 27 2023 *)

PARI
```
sqrt((5+sqrt(5))/2)
```

Equals 2*A019881. - Mohammed Yaseen, Nov 04 2020
Equals csc(A195693) = sec(A195723). - Amiram Eldar, May 28 2021
Equals i^(1/5) + i^(-1/5). - Gary W. Adamson, Jul 08 2022
Equals sqrt(2 + phi) = sqrt(A296184), with phi = A001622. - Wolfdieter Lang, Aug 28 2022
Equals Product_{k>=0} ((10*k + 2)*(10*k + 8))/((10*k + 1)*(10*k + 9)). - Antonio Graciá Llorente, Feb 24 2024
Equals Product_{k>=1} (1 - (-1)^k/A090771(k)). - Amiram Eldar, Nov 23 2024

A242671 Decimal expansion of k2, a Diophantine approximation constant such that the area of the "critical parallelogram" (in this case a square) is 4*k2.

Original entry on oeis.org

7, 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, 4

Offset: 0

Jean-François Alcover, May 20 2014

Quoting Steven Finch: "The slopes of the 'critical parallelogram' are (1+sqrt(5))/2 [phi] and (1-sqrt(5))/2 [-1/phi]."
Essentially the same as A229780, A134972, A134945, A098317 and A002163. - R. J. Mathar, May 23 2014
Let W_n be the collection of all binary words of length n that do not contain two consecutive 0's. Let r_n be the ratio of the total number of 1's in W_n divided by the total number of letters in W_n. Then lim_{n->oo} r_n = 0.723606... Equivalently, lim_{n->oo} A004798(n)/(n*A000045(n+2)) = 0.723606... - Geoffrey Critzer, Feb 04 2022
The limiting frequency of the digit 0 in the base phi representation of real numbers in the range [0,1], where phi is the golden ratio (A001622) (Rényi, 1957). - Amiram Eldar, Mar 18 2025

			k2 = 0.723606797749978969640917366873127623544...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.23, p. 176.

Alfréd Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar., Vol.8, No. 3-4 (1957), pp. 477-493.
Index entries for algebraic numbers, degree 2.

Cf. A000032, A000045, A001622, A002163, A004798, A020762, A081007, A094214, A094874, A098317, A134972, A229780, A244847, A296184, A300074, A344212.

RealDigits[(1+1/Sqrt[5])/2, 10, 100] // First

(1 + 1/sqrt(5))/2 \\ Stefano Spezia, Dec 07 2024

Equals (1 + 1/sqrt(5))/2.
Equals 1/A094874. - Michel Marcus, Dec 01 2018
From Amiram Eldar, Feb 11 2022: (Start)
Equals phi/sqrt(5), where phi is the golden ratio (A001622).
Equals lim_{k->oo} Fibonacci(k+1)/Lucas(k). (End)
From Amiram Eldar, Nov 28 2024: (Start)
Equals A344212/2 = A296184/5 = A300074^2 = sqrt(A229780).
Equals Product_{k>=1} (1 - 1/A081007(k)). (End)
Equals 1 - A244847. - Amiram Eldar, Mar 18 2025

A276886 Sums-complement of the Beatty sequence for 2 + phi.

Original entry on oeis.org

1, 2, 5, 6, 9, 12, 13, 16, 17, 20, 23, 24, 27, 30, 31, 34, 35, 38, 41, 42, 45, 46, 49, 52, 53, 56, 59, 60, 63, 64, 67, 70, 71, 74, 77, 78, 81, 82, 85, 88, 89, 92, 93, 96, 99, 100, 103, 106, 107, 110, 111, 114, 117, 118, 121, 122, 125, 128, 129, 132, 135, 136

Offset: 1

Clark Kimberling, Oct 01 2016

See A276871 for a definition of sums-complement and guide to related sequences.
From Michel Dekking, Apr 30 2019: (Start)
This sequence is a generalized Beatty sequence. According to Theorem 3.2 in the paper "The Frobenius problem for homomorphic embeddings of languages into the integers" this sequence (as a subset of the natural numbers) is the complement of the union of the two Beatty sequences
     V := A003231 and W = V+1 (as subsets of the natural numbers) given by
     V(n):= A(n)+2n = 3,7,10,14,..., W(n):=A(n)+2n+1 = 4,8,11,15,...
Here A = A000201, the lower Wythoff sequence.
Since the sequence Delta A = A014675 of first differences of A is the infinite Fibonacci word on the alphabet {2,1}, the sequence Delta V = (V(n+1)-V(n)) is the infinite Fibonacci word on the alphabet {4,3}. (Delta V equals A276867 shifted by 1.)
Now if for some k, Delta V(k) = 4, then a distance 3 plus a distance 1 are generated between three consecutive numbers in the complement, whereas if Delta V(k) = 3, then only a distance 3 is generated between two consecutive numbers in the complement.
This means that (skipping a(1)=1)
    Delta a = (a(n+1)-a(n)) = gamma(Delta V),
where gamma is the morphism
    gamma(4) = 31, gamma(3) = 3.
Since the Fibonacci word is a fixed point of the morphism 0->01, 1->0, this implies that Delta a, skipping a(1)=1, is the Fibonacci word on the alphabet {3,1}. It follows that
    a(n+1) = 2*A(n) - n + 1.
(End)

			The Beatty sequence for 2 + phi is 0 followed by A003231, which is (0,3,7,10,14,18,21,...), with difference sequence s = A276867 = (3,4,3,4,4,3,4,3,4,4,3,4,4,3,4,3,4,4,3,...). The sums s(j)+s(j+1)+...+s(k) include (3,4,7,8,10,12,14,15,...), with complement (1,2,5,6,9,12,13,16,...).

J.-P. Allouche, F. M. Dekking, Generalized Beatty sequences and complementary triples, arXiv:1809.03424 [math.NT], 2018.
Michel Dekking, The Frobenius problem for homomorphic embeddings of languages into the integers, Theoretical Computer Science 732, 7 July 2018, 73-79.
Index entries for sequences related to Beatty sequences

Cf. A003231, A276867, A276871, A296184(2+phi).

z = 500; r = 2 + GoldenRatio; b = Table[Floor[k*r], {k, 0, z}]; (* A003231 *)
t = Differences[b]; (* A276867 *)
c[k_, n_] := Sum[t[[i]], {i, n, n + k - 1}];
u[k_] := Union[Table[c[k, n], {n, 1, z - k + 1}]];
w = Flatten[Table[u[k], {k, 1, z}]]; Complement[Range[Max[w]], w];  (* A276886 *)

a(n) = 2*floor((n-1)*phi) - n + 2, where phi is the golden mean.

A309434 a(n) = floor(nIm(2e^(iPi/5))/(Im(2e^(i*Pi/5)) - 1)).

Original entry on oeis.org

6, 13, 20, 26, 33, 40, 46, 53, 60, 66, 73, 80, 87, 93, 100, 107, 113, 120, 127, 133, 140, 147, 154, 160, 167, 174, 180, 187, 194, 200, 207, 214, 220, 227, 234, 241, 247, 254, 261, 267, 274, 281, 287, 294, 301, 308, 314, 321, 328, 334, 341

Offset: 1

Karl V. Keller, Jr., Jun 06 2020

nonn

This is the Beatty sequence for Im(2*e^(i*Pi/5))/(Im(2*e^(i*Pi/5)) - 1).
This is the complement of A335137.
Im(2*e^(i*Pi/5))/(Im(2*e^(i*Pi/5)) - 1) = (5 + sqrt(5))/2 + sqrt(5 + 2*sqrt(5)) = 6.695717525925148250774877410... = 2 + phi + tan(2*Pi/5) = A296184 + A019970.
For n < 10, a(n) = A109235(n).
Re(2*e^(i*Pi/5))/(Re(2*e^(i*Pi/5)) - 1) = (3 + sqrt(5))/2 = 1 + phi = phi^2 = A104457.
Floor(n*Re(2*e^(i*Pi/5))/(Re(2*e^(i*Pi/5)) - 1)) is A001950 (floor(n*phi^2)).

			For n = 3, floor(3*6.69571) = 20.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences

Cf. A001950, A019970, A104457, A109235, A296184, A335137.

a[n_] := Floor[n * Im[2 * Exp[I * Pi/5]]/(Im[2 * Exp[I * Pi/5]] - 1)]; Array[a, 100] (* Amiram Eldar, Jul 06 2020 *)

from sympy import floor, im, exp, I, pi
for n in range(1, 101): print(floor(n*im(2*exp(I*pi/5))/(im(2*exp(I*pi/5)) - 1)), end=', ')

from sympy import floor, sqrt
for n in range(1, 101): print(floor(n*((5 + sqrt(5))/2 + sqrt(5 + 2*sqrt(5)))), end=', ')

cp's OEIS Frontend

A344476 Decimal expansion of the value of the Buchstab function at phi + 2 = (5 + sqrt(5))/2 (A296184).

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A104457 Decimal expansion of 1 + phi = phi^2 = (3 + sqrt(5))/2.

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A030191 Scaled Chebyshev U-polynomial evaluated at sqrt(5)/2.

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A116425 Decimal expansion of 2 + 2*cos(2*Pi/7).

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A332438 Decimal expansion of (2*cos(Pi/9))^2 = A332437^2.

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A351415 Intersection of Beatty sequences for (1+sqrt(5))/2 and sqrt(5).

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A188593 Decimal expansion of (diagonal)/(shortest side) of a golden rectangle.

A116425 Decimal expansion of 2 + 2cos(2Pi/7).

A309434 a(n) = floor(nIm(2e^(iPi/5))/(Im(2e^(i*Pi/5)) - 1)).