A242671 -id:A242671 - cp's OEIS frontend

A245278 Decimal expansion of k3, a Diophantine approximation constant such that the conjectured volume of the "critical parallelepiped" is 2^3*k3 (the 3-D analog of A242671).

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

Offset: 0

Jean-François Alcover, Jul 16 2014

			0.578416762788900759074602058146195674799483969436645501548317674941796...

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

Cf. A242671.

RealDigits[8/7*Cos[2*Pi/7]*Cos[Pi/7]^2, 10, 100] // First

8/7*cos(2*Pi/7)*cos(Pi/7)^2.

Also equals the positive root of 343*x^3 - 147*x^2 - 28*x - 1.

A182760 Beatty sequence for (3 + 5^(-1/2))/2.

Original entry on oeis.org

1, 3, 5, 6, 8, 10, 12, 13, 15, 17, 18, 20, 22, 24, 25, 27, 29, 31, 32, 34, 36, 37, 39, 41, 43, 44, 46, 48, 49, 51, 53, 55, 56, 58, 60, 62, 63, 65, 67, 68, 70, 72, 74, 75, 77, 79, 81, 82, 84, 86, 87, 89, 91, 93, 94, 96, 98, 99, 101, 103, 105, 106, 108, 110, 112, 113, 115, 117, 118, 120, 122, 124, 125, 127, 129, 130, 132, 134, 136, 137, 139, 141, 143, 144, 146, 148, 149, 151, 153, 155, 156, 158, 160, 162, 163, 165, 167, 168

Offset: 1

Clark Kimberling, Nov 28 2010

Suppose that u and v are positive real numbers for which the sets S(u)={j*u} and S(v)={k*v}, for j>=1 and k>=1, are disjoint. Let a(n) be the position of n*u when the numbers in S(u) and S(v) are jointly ranked. Then, as is easy to prove, a is the Beatty sequence of the number r=1+u/v, and the complement of a is the Beatty sequence of s=1+v/u. For A182760, take u = golden ratio = (1+sqrt(5))/2 and v=sqrt(5), so that r=(3+5^(-1/2))/2 and s=(7-sqrt(5))/2.

			Let u=(1+sqrt(5))/2 and v=sqrt(5).  When the numbers ju and kv are jointly ranked, we write U for numbers of the form ju and V for the others.  Then the ordering of the ranked numbers is given by U V U V U U V U V U V U U ..  The positions of U are given by A182760.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Cf. A182761 (the complement of A182760), A242671

[Floor(n*(3+5^(-1/2))/2): n in [1..70]]; // Vincenzo Librandi, Oct 25 2011

Table[Floor[Sqrt[n/20]+3*n/2], {n,1,100}] (* G. C. Greubel, Jan 11 2018 *)

a(n)=floor(sqrt(n/20)+3*n/2) \\ Charles R Greathouse IV, Jul 02 2013

a(n) = floor(r*n), where r = (3 + 5^(-1/2))/2 = 1.72360...

More than the usual number of terms are shown in order to distinguish this from a very similar sequence. - N. J. A. Sloane, Jan 20 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)

A244847 Decimal expansion of rho_c = (5-sqrt(5))/10, the asymptotic critical density for the hard hexagon model.

Original entry on oeis.org

2, 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

Offset: 0

Jean-François Alcover, Nov 12 2014

The vertical distance between the accumulation point and the outermost point of a golden spiral inscribed inside a golden rectangle with dimensions phi and 1 along the x and y axes, respectively (the horizontal distance is A176015). - Amiram Eldar, May 18 2021

The limiting frequency of the digit 1 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

			0.2763932022500210303590826331268723764559381640388474275729102754589479...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.2 The Golden Mean, phi, p. 7.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.12.1 Phase transitions in Lattice Gas Models, p. 347.

Rodney J. Baxter Hard hexagons: exact solution, Journal of Physics A: Mathematical and General, Vol. 13, No. 3 (1980), pp. L61-L70, alternative link.
P. S. Bruckman and I. J. Good, A Generalization of a Series of De Morgan with Applications of Fibonacci Type, The Fibonacci Quarterly, Vol. 14, No. 3 (1976), pp. 193-196.
Alfréd Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar., Vol.8, No. 3-4 (1957), pp. 477-493.
Eric Weisstein's MathWorld, Hard Hexagon Entropy Constant.
Wikipedia, Hard Hexagon Model.
Index entries for algebraic numbers, degree 2.

Cf. A242671, A244593.
Cf. A000032, A000045.
Essentially the same sequence of digits as A229760 and A187799.

RealDigits[(5 - Sqrt[5])/10, 10, 102] // First

Equals 1/(sqrt(5)*phi), where phi = (1+sqrt(5))/2 = A001622. - Vaclav Kotesovec, Nov 13 2014
Equals lim_{n -> infinity} A000045(n)/A000032(n+1). - Bruno Berselli, Jan 22 2018
Equals Sum_{n>=1} A000045(3^(n-1))/A000032(3^n) = Sum_{n>=1} A045529(n-1)/A006267(n). - Amiram Eldar, Dec 20 2018
Equals 1 - A242671. - Amiram Eldar, Mar 18 2025

A081007 a(n) = Fibonacci(4n+1) - 1, or Fibonacci(2n)*Lucas(2n+1).

Original entry on oeis.org

0, 4, 33, 232, 1596, 10945, 75024, 514228, 3524577, 24157816, 165580140, 1134903169, 7778742048, 53316291172, 365435296161, 2504730781960, 17167680177564, 117669030460993, 806515533049392, 5527939700884756, 37889062373143905, 259695496911122584

Offset: 0

R. K. Guy, Mar 01 2003

Also the index of the first of two consecutive triangular numbers whose sum is equal to the sum of two consecutive heptagonal numbers. - Colin Barker, Dec 20 2014

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..500
Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

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

Cf. A081018, A242671.

List([0..30], n-> Fibonacci(4*n+1)-1); # G. C. Greubel, Jul 14 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

Table[Fibonacci[4n+1] -1, {n,0,30}] (* Wesley Ivan Hurt, Oct 06 2013 *)
LinearRecurrence[{8,-8,1},{0,4,33},30] (* Harvey P. Dale, Jul 31 2018 *)
Table[Fibonacci[2n]LucasL[2n+1], {n,0,30}] (* Rigoberto Florez, Apr 19 2019 *)

A081007(n):=fib(4*n+1)-1$
makelist(A081007(n),n,0,30); /* Martin Ettl, Nov 12 2012 */

concat(0, Vec(x*(4+x)/((1-x)*(1-7*x+x^2)) + O(x^30))) \\ Colin Barker, Dec 20 2014

vector(30, n, n--; fibonacci(4*n+1)-1) \\ G. C. Greubel, Jul 14 2019

[fibonacci(4*n+1)-1 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.: x*(4+x)/((1-x)*(1-7*x+x^2)). - Colin Barker, Jun 24 2012
a(n) = Sum_{i=1..2n} binomial(2n+i, 2n-i). - Wesley Ivan Hurt, Oct 06 2013
a(n) = Sum_{i=0..2n-1} F(i)*L(i+2), F(i) = A000045(i) and L(i) = A000032(i). - Rigoberto Florez, Apr 19 2019
Product_{n>=1} (1 - 1/a(n)) = (1 + 1/sqrt(5))/2 (A242671). - Amiram Eldar, Nov 28 2024

More terms from James Sellers, Mar 03 2003

A377995 Decimal expansion of the dihedral angle, in radians, between square and pentagonal faces in a (small) rhombicosidodecahedron.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Nov 15 2024

Also the dihedral angle, in radians, between square and 10-gonal faces in a truncated icosidodecahedron (great rhombicosidodecahedron).

			2.588018294692747986954110653190234364162145576674...

Eric Weisstein's World of Mathematics, Great Rhombicosidodecahedron.
Eric Weisstein's World of Mathematics, Small Rhombicosidodecahedron.

Cf. A242671, A377996.

First[RealDigits[ArcCos[-Sqrt[(5 + Sqrt[5])/10]], 10, 100]] (* or *)
First[RealDigits[Min[PolyhedronData["Rhombicosidodecahedron", "DihedralAngles"]], 10, 100]]

Equals arccos(-sqrt((5 + sqrt(5))/10)) = arccos(-sqrt(A242671)).

A049658 a(n) = (F(8*n+5) - 2)/3, where F = A000045 (the Fibonacci sequence).

Original entry on oeis.org

1, 77, 3648, 171409, 8052605, 378301056, 17772097057, 834910260653, 39223010153664, 1842646566961585, 86565165637040861, 4066720138373958912, 191049281337939028033, 8975249502744760358669

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..595
Index entries for linear recurrences with constant coefficients, signature (48, -48, 1).

Cf. A000045, A242671.

[(Fibonacci(8*n+5) - 2)/3: n in [0..30]]; // G. C. Greubel, Dec 02 2017

(Fibonacci[8Range[0,20]+5]-2)/3 (* or *) LinearRecurrence[{48,-48,1},{1,77,3648},20] (* Harvey P. Dale, Jun 20 2013 *)

for(n=0,30, print1((fibonacci(8*n+5) - 2)/3, ", ")) \\ G. C. Greubel, Dec 02 2017

G.f.: (1+29*x)/(1-48*x+48*x^2-x^3).
a(0)=1, a(1)=77, a(2)=3648, a(n) = 48*a(n-1)-48*a(n-2)+a(n-3). - Harvey P. Dale, Jun 20 2013
Product_{n>=1} (1 - 1/a(n)) = 3*(5+sqrt(5))/22 = (15/11) * A242671. - Amiram Eldar, Nov 28 2024

Description corrected by and more terms from Michael Somos

A322258 Decimal expansion of exp(-phi/sqrt(5)), where phi is the golden ratio.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Dec 01 2018

			0.48499980129295802523175132230095248348065996564155...

J. Sandor and B. Crstici, Handbook of Number Theory II, Springer, 2004, pp. 54-55, p. 182.

Don Redmond, Infinite products and Fibonacci numbers, Fib. Quart., Vol. 32, No. 3 (1994), pp. 234-239.
Index entries for transcendental numbers

Cf. A000010, A000032, A000045, A001622, A242671, A322259.

RealDigits[Exp[-GoldenRatio/Sqrt[5]], 10, 120][[1]]

exp(-(1+1/sqrt(5))/2) \\ Charles R Greathouse IV, Nov 21 2024

Equals Product_{k>=1} (L(k)/(sqrt(5)*F(k)))^(phi(k)/k), where L(k) and F(k) are the Lucas and Fibonacci numbers, and phi(k) is the Euler totient function.

Equals exp(-A242671).

cp's OEIS Frontend

A245278 Decimal expansion of k3, a Diophantine approximation constant such that the conjectured volume of the "critical parallelepiped" is 2^3*k3 (the 3-D analog of A242671).

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A182760 Beatty sequence for (3 + 5^(-1/2))/2.

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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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A244847 Decimal expansion of rho_c = (5-sqrt(5))/10, the asymptotic critical density for the hard hexagon model.

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A081007 a(n) = Fibonacci(4n+1) - 1, or Fibonacci(2n)*Lucas(2n+1).

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