A002390 -id:A002390 - cp's OEIS frontend

A371529 Decimal expansion of Product_{k>=2} (1 + (-1)^k/Lucas(k)).

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

Offset: 1

Amiram Eldar, Mar 26 2024

			1.07248271775513062588537881652660869304392049333099...

Daniel Duverney, Carsten Elsner, Masanobu Kaneko, and Yohei Tachiya, A criterion of algebraic independence of values of modular functions and an application to infinite products involving Fibonacci and Lucas numbers, Research in Number Theory, Vol. 8 (2022), Article 31; alternative link.
Eric Weisstein's World of Mathematics, Dedekind Eta Function.
Wikipedia, Dedekind eta function.

Cf. A000032, A001622, A002390, A337668, A337669, A371525, A371526, A371527, A371528, A371530.

With[{eta = DedekindEta, tau0 = Log[GoldenRatio]*I/Pi}, RealDigits[(Surd[GoldenRatio, 4] / Sqrt[5]) * eta[2*tau0]^3 * eta[3*tau0] / (eta[tau0]^2 * eta[4*tau0]), 10, 120][[1]]]

prodinf(k = 2, 1 + (-1)^k/(fibonacci(k-1) + fibonacci(k+1)))

Equals Product_{k>=2} (1 + (-1)^k/A000032(k)).
Equals A371525 / (2*sqrt(5)).
Equals (phi^(1/4) / sqrt(5)) * eta(2*tau_0)^3 * eta(3*tau_0) / (eta(tau_0)^2 * eta(4*tau_0)), where phi is the golden ratio (A001622), tau_0 = i*log(phi)/Pi, and i = sqrt(-1) (Duverney et al., 2022).

A371530 Decimal expansion of Product_{k>=1} (1 - (-1)^k/Lucas(k)).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Mar 26 2024

			1.50218004433245676912076258176556998802715258088888...

Daniel Duverney, Carsten Elsner, Masanobu Kaneko, and Yohei Tachiya, A criterion of algebraic independence of values of modular functions and an application to infinite products involving Fibonacci and Lucas numbers, Research in Number Theory, Vol. 8 (2022), Article 31; alternative link.
Eric Weisstein's World of Mathematics, Dedekind Eta Function.
Wikipedia, Dedekind eta function.

Cf. A000032, A001622, A002390, A337668, A337669, A371525, A371526, A371527, A371528, A371529.

With[{eta = DedekindEta, tau0 = Log[GoldenRatio]*I/Pi}, RealDigits[2 * Surd[GoldenRatio, 4] * eta[2*tau0]^2 * eta[6*tau0]/(eta[3*tau0] * eta[4*tau0]), 10, 120][[1]]]

prodinf(k = 1, 1 - (-1)^k/(fibonacci(k-1) + fibonacci(k+1)))

Equals Product_{k>=2} (1 - (-1)^k/A000032(k)).
Equals (2*sqrt(5)) * A371526.
Equals 2 * phi^(1/4) * eta(2*tau_0)^2 * eta(6*tau_0) / (eta(3*tau_0) * eta(4*tau_0)), where phi is the golden ratio (A001622), tau_0 = i*log(phi)/Pi, and i = sqrt(-1) (Duverney et al., 2022).

A139347 Decimal expansion of negated tangent of the golden ratio. That is, the decimal expansion of -tan((1+sqrt(5))/2).

Original entry on oeis.org

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

Offset: 2

Mohammad K. Azarian, Apr 15 2008

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 13 2019

			-21.15380078249327464858628117032582559788124367464826...

Index entries for transcendental numbers.

Cf. A001622, A094214, A104457, A098317, A002390, A139339, A139340, A139341, A139342, A139345, A139346, A139348.

RealDigits[Tan[GoldenRatio], 10, 100][[1]] (* Amiram Eldar, Feb 07 2022 *)

-tan((sqrt(5)+1)/2) \\ Charles R Greathouse IV, May 13 2019

Equals tan(A001622).
From Amiram Eldar, Feb 07 2022: (Start)
Equals 1/A139348.
Equals A139345/A139346. (End)

Offset corrected by Mohammad K. Azarian, Dec 13 2008

Sign added to definition by R. J. Mathar, Feb 05 2009

A139348 Decimal expansion of negated cotangent of the golden ratio. That is, the decimal expansion of -cot((1+sqrt(5))/2).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Apr 15 2008

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 13 2019

			0.04727282866479448118935650960621633420056105722556...

Index entries for transcendental numbers.

Cf. A001622, A094214, A104457, A098317, A002390, A139339, A139340, A139341, A139342, A139345, A139346, A139347.

Join[{0},RealDigits[-Cot[GoldenRatio],10,120][[1]]] (* Harvey P. Dale, Sep 18 2013 *)

-cotan((sqrt(5)+1)/2) \\ Charles R Greathouse IV, May 13 2019

Equals cot(A001622).
From Amiram Eldar, Feb 07 2022: (Start)
Equals 1/A139347.
Equals A139346/A139345. (End)

Added sign in definition. Leading zero dropped by R. J. Mathar, Feb 05 2009

A202541 Decimal expansion of the number x satisfying e^(2x) - e^(-2x) = 1.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Dec 21 2011

See A202537 for a guide to related sequences. The Mathematica program includes a graph.

Archimedes's-like scheme: set p(0) = 1/(2*sqrt(5)), q(0) = 1/4; p(n+1) = 2*p(n)*q(n)/(p(n)+q(n)) (arithmetic mean of reciprocals, i.e., 1/p(n+1) = (1/p(n) + 1/q(n))/2), q(n+1) = sqrt(p(n+1)*q(n)) (geometric mean, i.e., log(q(n+1)) = (log(p(n+1)) + log(q(n)))/2), for n >= 0. The error of p(n) and q(n) decreases by a factor of approximately 4 each iteration, i.e., approximately 2 bits are gained by each iteration. Set r(n) = (2*q(n) + p(n))/3, the error decreases by a factor of approximately 16 for each iteration, i.e., approximately 4 bits are gained by each iteration. For a similar scheme see also A244644. - A.H.M. Smeets, Jul 12 2018

			0.24060591252980172374887945671218421156759216719...

R. S. Melham and A. G. Shannon, Inverse trigonometric and hyperbolic summation formulas involving generalized Fibonacci numbers, Fibonacci Quarterly, Vol. 33, No. 1 (1995), pp. 32-40.
D. Zagier, Algebraic numbers close to both 0 and 1, Mathematics of Computation, Vol. 61, No. 203 (1993), pp. 485-491.
Index entries for transcendental numbers

Cf. A001622, A002390, A202537.

u = 2; v = 2;
f[x_] := E^(u*x) - E^(-v*x); g[x_] := 1
Plot[{f[x], g[x]}, {x, -2, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .2, .3}, WorkingPrecision -> 110]
RealDigits[r]   (* A202541 *)
RealDigits[ Log[ (1+Sqrt[5])/2 ] / 2, 10, 99] // First (* Jean-François Alcover, Feb 27 2013 *)
RealDigits[ FindRoot[ Exp[2x] - Exp[-2x] == 1, {x, 1}, WorkingPrecision -> 128][[1, 2]], 10, 111][[1]] (* Robert G. Wilson v, Jul 23 2018 *)

asinh(1/2)/2 \\ Michel Marcus, Jul 12 2018

Equals (1/2)*arcsinh(1/2) or (1/2)*log(phi), phi being the golden ratio. - A.H.M. Smeets, Jul 12 2018
Equals Sum_{k>=1} (-1)^(k+1) * arctanh(1/Fibonacci(3*k)^2) (Melham and Shannon, 1995). - Amiram Eldar, Oct 04 2021
Equals A002390/2. - Alois P. Heinz, Jul 14 2022
Equals arctanh(sqrt(5)-2). - Amiram Eldar, Feb 09 2024

A349850 Decimal expansion of Sum_{k>=1} H(k)*F(k)/2^k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number and F(k) = A000045(k) is the k-th Fibonacci number.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Dec 02 2021

			3.96874800690391485217106364061998568869842436396224...

Hideyuki Ohtsuka, Problem B-1200, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 54, No. 4 (2016), p. 367; Harmonic and Fiboancci [sic]/Lucas Numbers, Solution to Problem B-1200 by Kenny B. Davenport, ibid., Vol. 55, No. 4 (2017), pp. 372-373.

Cf. A000045, A001008, A001622, A002162, A002163, A002390, A002805, A349851.

RealDigits[2*Log[2] + 12*Log[GoldenRatio]/Sqrt[5], 10, 100][[1]]

Equals log(4*phi^(12/sqrt(5))) = 2*log(2) + 12*log(phi)/sqrt(5), where phi is the golden ratio (A001622).

A385445 Decimal expansion of (-1 + 3phi)sqrt(3 - phi), with the golden section phi = A001622.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Jul 01 2025

This constant c gives the real part of -2*11*Z = (c + d*i), where Z is the (finite) fixed point of a complex function w (of the loxodromic type) mapping iteratively the vertices of golden triangles, starting with vertices (D_1, D_2, D_3), circumscribed by the unit circle with center at the origin, and D_1 = i, D_2 = (s - phi*i)/2 and D_3 = (-s - phi*i)/2. This function is w(z) = a*z + b, with a = (-1 + phi) * exp(-(3*Pi/5)*i) = -((2 - phi) + s*i)/2 and b = (1 - phi)*i, where s = sqrt(3 - phi) = A182007 (the length of the base (D_2, D_3) of the first triangle, also called s_1).
The imaginary part of -2*11*Z is d = -7 + 10*phi = A385446.
If the fixed point Z = -(0.20594... + 0.41728...*i) is chosen as origin then the loxodromic map is W(z') = a*z' (where z' = z - Z and W(z') = w(z'+Z) - Z).
For details see the linked paper, eqs.(5a,b) for w(z), eq.(6) for Z and eq.(7) for W(z'). (In eq.(5b) the i is missing in the exponent.) The nesting of golden triangles as shown in Fig. 1 of the link leads to the fixed point Z.
The vertices of the nested golden triangles can be connected by a spiral built of circular sections with angle 108 degrees, centered at vertices D_{n+3} and shrinking radii s_n =(- 1 + phi)^(n-1)*s. Note that the curvature of this spiral is not continuous.
The length(Z, D_n) =: rho_n of the spokes of the spiral is (-1 + phi)^(n-1)*rho_1, with sqrt(11)*rho_1 = sqrt(8 + 9*phi) = sqrt(5 + 7*phi)*s = A385447.
For the length ratio rho_1/s see A385448.
The logarithmic spiral connecting the vertices D_{n+1} is given in polar coordinates by rho(Phi) = rho_1 * exp((-(5/(3*Pi)) * log(phi)*Phi), with the vertices obtained in polar coordinates for Phi = Phi_n = (3*Pi/5)*n, namely rho(Phi_n) = rho_{n+1}, for n >= 0. For log(phi) see A002390. Note that the nonnegative x-axis is now along Z, D_1. The angle(Z, D1, D4) =: gamma is given by arctan((18 - 11*phi)/s) = arcsin(rho_4 / 2) = 0.169860704... or 9.7323... degrees. See Fig. 3 of the linked paper.

			4.5307685931859751743612240909981473232388869294682093...

Wolfdieter Lang, On a Conformal Mapping of Golden Triangles, Papua New Guinea Journal of Maths. 1991, Vol.2, No:2, pp. 12 - 18 (with corrections).

Cf. A001622, A002390, A090550, A182007, A385446, A385447, A385448.

RealDigits[Sqrt[27 - 4*GoldenRatio], 10, 120][[1]] (* Amiram Eldar, Jul 02 2025 *)

Equals (-1 + 3*phi)*sqrt(3 - phi) = (A090550 - 2)*A182007.

Equals sqrt(27 - 4*phi).

A072876 a(1) = a(2) = a(3) = a(4) = 1 and a(n) = (a(n-1)*a(n-3) + a(n-2)^3)/a(n-4) for n >= 5.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 11, 49, 739, 41926, 36876163, 1504578225617, 67856786028033600651, 81238311359334144709516343054051, 8472940010945536421401513734595877223414710434640386

Offset: 1

Benoit Cloitre, Jul 28 2002

A variation of a Somos-4 sequence with a(n-2)^3 in place of a(n-2)^2.

Seiichi Manyama, Table of n, a(n) for n = 1..21

Cf. A002390, A006720, A072877, A111459.

Nest[Append[#, (#[[-1]]*#[[-3]] + #[[-2]]^3)/#[[-4]] ] &, {1, 1, 1, 1}, 11] (* Michael De Vlieger, May 12 2019 *)
RecurrenceTable[{a[1]==a[2]==a[3]==a[4]==1,a[n]==(a[n-1]a[n-3]+a[n-2]^3)/ a[n-4]},a,{n,20}] (* Harvey P. Dale, May 15 2019 *)

Lim_{n->infinity} (log(log a(n)))/n = log((1+sqrt(5))/2) or about 0.48. See A002390. However, convergence is extremely slow. - Andrew Hone, Nov 15 2005
From Jon E. Schoenfield, May 12 2019: (Start)
It appears that, for n >= 1,
  a(n) = ceiling(e^(c0*phi^n + d0/phi^n)) if n is even,
         ceiling(e^(c1*phi^n + d1/phi^n)) if n is odd,
where
  phi = (1 + sqrt(5))/2,
   c0 =  0.087172479898911051233710515749226588954735607680...
   c1 =  0.087662681482404614007222542134598226046349621976...
   d0 = -9.574280373370101810186207466479291633433387765559...
   d1 = -4.425515288739040257644546086989175506652492968654...
(End)

A129187 Decimal expansion of arcsinh(1/3).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Jul 27 2008

Archimedes's-like scheme: set p(0) = 1/sqrt(10), q(0) = 1/3; p(n+1) = 2*p(n)*q(n)/(p(n)+q(n)) (arithmetic mean of reciprocals, i.e., 1/p(n+1) = (1/p(n) + 1/q(n))/2), q(n+1) = sqrt(p(n+1)*q(n)) (geometric mean, i.e., log(q(n+1)) = (log(p(n+1)) + log(q(n)))/2), for n >= 0. The error of p(n) and q(n) decreases by a factor of approximately 4 each iteration, i.e., approximately 2 bits are gained by each iteration. Set r(n) = (2*q(n) + p(n))/3, the error decreases by a factor of approximately 16 for each iteration, i.e., approximately 4 bits are gained by each iteration. For a similar scheme see also A244644. - A.H.M. Smeets, Jul 12 2018

			0.32745015023725844332253525998825812770052452899076745127562...

Muniru A Asiru, Table of n, a(n) for n = 0..2000
Index entries for transcendental numbers

Cf. A002390, A129200, A129269, A244644.

RealDigits[ArcSinh[1/3], 10, 111][[1]] (* Robert G. Wilson v, Jul 23 2018 *)

asinh(1/3) \\ Charles R Greathouse IV, Mar 25 2014

Equals log((1 + sqrt(10))/3). - Jianing Song, Jul 12 2018

Equals arccoth(sqrt(10)). - Amiram Eldar, Feb 09 2024

A139349 Decimal expansion of negated secant of the golden ratio. That is, the decimal expansion of -sec((1+sqrt(5))/2).

Original entry on oeis.org

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

Offset: 2

Mohammad K. Azarian, Apr 15 2008

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 13 2019

			21.17742400636614440872804040937130213307185355364174...

Index entries for transcendental numbers.

Cf. A001622, A094214, A104457, A098317, A002390, A139339, A139340, A139341, A139342, A139345, A139346, A139347, A139348.

RealDigits[-Sec[GoldenRatio],10,120][[1]] (* Harvey P. Dale, Dec 08 2011 *)

-1/cos((sqrt(5)+1)/2) \\ Charles R Greathouse IV, May 13 2019

Equals sec(A001622).

Equals 1/A139346. - Amiram Eldar, Feb 07 2022

Offset corrected by Mohammad K. Azarian, Dec 13 2008

Sign in definition added by R. J. Mathar, Feb 05 2009

cp's OEIS Frontend

A371529 Decimal expansion of Product_{k>=2} (1 + (-1)^k/Lucas(k)).

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A371530 Decimal expansion of Product_{k>=1} (1 - (-1)^k/Lucas(k)).

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A139347 Decimal expansion of negated tangent of the golden ratio. That is, the decimal expansion of -tan((1+sqrt(5))/2).

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A139348 Decimal expansion of negated cotangent of the golden ratio. That is, the decimal expansion of -cot((1+sqrt(5))/2).

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A202541 Decimal expansion of the number x satisfying e^(2x) - e^(-2x) = 1.

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A349850 Decimal expansion of Sum_{k>=1} H(k)*F(k)/2^k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number and F(k) = A000045(k) is the k-th Fibonacci number.

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A385445 Decimal expansion of (-1 + 3*phi)*sqrt(3 - phi), with the golden section phi = A001622.

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A385445 Decimal expansion of (-1 + 3phi)sqrt(3 - phi), with the golden section phi = A001622.