A075778 -id:A075778 - cp's OEIS frontend

A060006 Decimal expansion of real root of x^3 - x - 1 (the plastic constant).

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

Offset: 1

Fabian Rothelius, Mar 14 2001

Has been also called the silver number, also the plastic number.
This is the smallest Pisot-Vijayaraghavan number.
The name "plastic number" goes back to the Dutch Benedictine monk and architect Dom Hans van der Laan, who gave this name 4 years after the discovery of the number by the French engineer Gérard Cordonnier in 1924, who used the name "radiant number". - Hugo Pfoertner, Oct 07 2018
Sometimes denoted by the symbol rho. - Ed Pegg Jr, Feb 01 2019
Also the solution of 1/x + 1/(1+x+x^2) = 1. - Clark Kimberling, Jan 02 2020
Given any complex p such that real(p)>-1, this constant is the only real solution of the equation z^p+z^(p+1)=z^(p+3), and the only attractor of the complex mapping z->M(z,p), where M(z,p)=(z^p+z^(p+1))^(1/(p+3)), convergent from any complex plane point. - Stanislav Sykora, Oct 14 2021
The Pisot-Vijayaraghavan numbers were named after the French mathematician Charles Pisot (1910-1984) and the Indian mathematician Tirukkannapuram Vijayaraghavan (1902-1955). - Amiram Eldar, Apr 02 2022
The sequence a(n) = v_3^floor(n^2/4) where v_n is the smallest, positive, real solution to the equation (v_n)^n = v_n + 1 satisfies the Somos-5 recursion a(n+3)*a(n-2) = a(n+2)*a(n-1) + a(n+1)*a(n) for all n in Z. Also true if floor is removed. - Michael Somos, Mar 24 2023

			1.32471795724474602596090885447809734...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.2.
Midhat J. Gazalé, Gnomon: From Pharaohs to Fractals, Princeton University Press, Princeton, NJ, 1999, see Chap. VII.
Donald E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4, p. 236.
Ian Stewart, A Guide to Computer Dating (Feedback), Scientific American, Vol. 275 No. 5, November 1996, p. 118.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Alex Bellos, The golden ratio has spawned a beautiful new curve: the Harriss spiral, The Guardian, Jan 13 2015.
Gamaliel Cerda-Morales, New Identities for Padovan Numbers, arXiv:1904.05492 [math.CO], 2019.
Brady Haran and Edmund Harriss, The Plastic Ratio, Numberphile video (2019).
Ed Pegg Jr., Pictures based on the plastic constant
Simon Plouffe, Smallest Pisot-Vijayaraghavan number to 50000 digits
Simon Plouffe, The Smallest Pisot-Vijayaraghavan number
F. Rothelius, Formulae.
Ian Stewart, Tales of a Neglected Number, Mathematical Recreations, Scientific American, Vol. 274, No. 6 (1996), pp. 102-103.
Ian Stewart, Tales of a Neglected Number, Mathematical Recreations, Scientific American, Vol. 274, No. 6 (1996), pp. 102-103.
Dom Hans van der Laan, Le nombre plastique: Quinze leçons sur l’ordonnance architectonique, Brill Academic Pub., Leiden, 1960.
Michel Waldschmidt, Lectures on Multiple Zeta Values, IMSC 2011.
Eric Weisstein's World of Mathematics, Maverick Graph.
Eric Weisstein's World of Mathematics, Pisot Number.
Eric Weisstein's World of Mathematics, Plastic Constant.
Wikipedia, Plastic number.
Index entries for algebraic numbers, degree 3

Cf. A001622. A072117 gives continued fraction.
Cf. A006888, A010527, A051016, A051017, A084252, A075778 (inverse), A126772.
Other Pisot numbers: A086106, A092526, A228777, A293506, A293508, A293509, A293557.
Cf. A002620, A006721, A134816.

SetDefaultRealField(RealField(100)); ((3+Sqrt(23/3))/6)^(1/3) + ((3-Sqrt(23/3))/6)^(1/3); // G. C. Greubel, Mar 15 2019

(1/2 +sqrt(23/3)/6)^(1/3) + (1/2-sqrt(23/3)/6)^(1/3) ; evalf(%,130) ; # R. J. Mathar, Jan 22 2013

RealDigits[ Solve[x^3 - x - 1 == 0, x][[1, 1, 2]], 10, 111][[1]] (* Robert G. Wilson v, Sep 30 2009 *)
s = Sqrt[23/108]; RealDigits[(1/2 + s)^(1/3) + (1/2 - s)^(1/3), 10, 111][[1]] (* Robert G. Wilson v, Dec 12 2017 *)
RealDigits[Root[x^3-x-1,1],10,120][[1]] (* or *) RealDigits[(Surd[9-Sqrt[69],3]+Surd[9+Sqrt[69],3])/(Surd[2,3]Surd[9,3]),10,120][[1]] (* Harvey P. Dale, Sep 04 2018 *)

allocatemem(932245000); default(realprecision, 20080); x=solve(x=1, 2, x^3 - x - 1); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b060006.txt", n, " ", d)); \\ Harry J. Smith, Jul 01 2009

(1/2 +sqrt(23/3)/6)^(1/3) + (1/2-sqrt(23/3)/6)^(1/3) \\ Altug Alkan, Apr 10 2016

polrootsreal(x^3-x-1)[1] \\ Charles R Greathouse IV, Aug 28 2016

default(realprecision, 110); digits(floor(solve(x=1, 2, x^3 - x - 1)*10^105)) /* Michael Somos, Mar 24 2023 */

numerical_approx(((3+sqrt(23/3))/6)^(1/3) + ((3-sqrt(23/3))/6)^(1/3), digits=100) # G. C. Greubel, Mar 15 2019

Equals (1/2+sqrt(23/108))^(1/3) + (1/2-sqrt(23/108))^(1/3). - Henry Bottomley, May 22 2003
Equals CubeRoot(1 + CubeRoot(1 + CubeRoot(1 + CubeRoot(1 + ...)))). - Gerald McGarvey, Nov 26 2004
Equals sqrt(1+1/sqrt(1+1/sqrt(1+1/sqrt(1+...)))). - Gerald McGarvey, Mar 18 2006
Equals (1/2 +sqrt(23/3)/6)^(1/3) + (1/2-sqrt(23/3)/6)^(1/3). - Eric Desbiaux, Oct 17 2008
Equals Sum_{k >= 0} 27^(-k)/k!*(Gamma(2*k+1/3)/(9*Gamma(k+4/3)) - Gamma(2*k-1/3)/(3*Gamma(k+2/3))). - Robert Israel, Jan 13 2015
Equals sqrt(Phi) = sqrt(1.754877666246...) (see A109134). - Philippe Deléham, Sep 29 2020
Equals cosh(arccosh(3*c)/3)/c, where c = sqrt(3)/2 (A010527). - Amiram Eldar, May 15 2021
Equals 1/hypergeom([1/5, 2/5, 3/5, 4/5], [2/4, 3/4, 5/4], -5^5/4^4). - Gerry Martens, Mar 16 2025

Edited and extended by Robert G. Wilson v, Aug 03 2002

Removed incorrect comments, Joerg Arndt, Apr 10 2016

A092526 Decimal expansion of (2/3)cos( (1/3)arccos(29/2) ) + 1/3, the real root of x^3 - x^2 - 1.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Apr 07 2004

This is the limit x of the ratio N(n+1)/N(n) for n -> infinity of the Narayana sequence N(n) = A000930(n). The real root of x^3 - x^2 - 1. See the formula section. - Wolfdieter Lang, Apr 24 2015
This is the fourth smallest Pisot number. - Iain Fox, Oct 13 2017
Sometimes called the supergolden ratio or Narayana's cows constant, and denoted by the symbol psi. - Ed Pegg Jr, Feb 01 2019

			1.46557123187676802665673122521993910802557756847228570164318311124926...

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.3.
Paul J. Nahin, The Logician and the Engineer, How George Boole and Claude Shannon Created the Information Age, Princeton University Press, Princeton and Oxford, 2013, Chap. 7: Some Combinational Logic Examples, Section 7.1: Channel Capacity, Shannon's Theorem, and Error-Detection Theory, page 120.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Simon Baker, Exceptional digit frequencies and expansions in non-integer bases, arXiv:1711.10397 [math.DS], 2017. See the beta(2) constant pp. 3-4.
H. R. P. Ferguson, On a Generalization of the Fibonacci Numbers Useful in Memory Allocation Schema or All About the Zeroes of Z^k - Z^{k - 1} - 1, k > 0, Fibonacci Quarterly, Volume 14, Number 3, October, 1976 (see Table 2, p. 236).
Ed Pegg Jr., Images based on the supergolden ratio
Michael Penn, What is the super-golden ratio??, YouTube video, 2022.
Mario Raso, Integer Sequences in Cryptography: A New Generalized Family and its Application, Ph. D. Thesis, Sapienza University of Rome (Italy 2025). See p. 59.
Eric Weisstein's World of Mathematics, Pisot Number.
Eric Weisstein's World of Mathematics, Supergolden Ratio.
Wikipedia, Pisot number
Wikipedia, Supergolden ratio
Index entries for algebraic numbers, degree 3

Cf. A088559, A075778, A076725, A000930, A263719.
Other Pisot numbers: A060006, A086106, A228777, A293506, A293508, A293509, A293557.
Cf. A381124 (numerators of convergents).
Cf. A381125 (denominators of convergents).

RealDigits[(2 Cos[ ArcCos[ 29/2]/3] + 1)/3, 10, 111][[1]] (* Robert G. Wilson v, Apr 12 2004 *)
RealDigits[ Solve[ x^3 - x^2 - 1 == 0, x][[1, 1, 2]], 10, 111][[1]] (* Robert G. Wilson v, Oct 10 2013 *)

allocatemem(932245000); default(realprecision, 20080); x=solve(x=1, 2, x^3 - x^2 - 1); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b092526.txt", n, " ", d));  \\ Harry J. Smith, Jun 21 2009

The real root of x^3 - x^2 - 1. - Franklin T. Adams-Watters, Oct 12 2006
The only real irrational root of x^4-x^2-x-1 (-1 is also a root). [Nahim]
Equals (2/3)*cos( (1/3)*arccos(29/2) ) + 1/3.
Equals 1 + A088559.
Equals (1/6)*(116+12*sqrt(93))^(1/3) + 2/(3*(116+12*sqrt(93))^(1/3)) + 1/3. - Vaclav Kotesovec, Dec 18 2014
Equals 1/A263719. - Alois P. Heinz, Apr 15 2018
Equals (1 + 1/r + r)/3 where r = ((29 + sqrt(837))/2)^(1/3). - Peter Luschny, Apr 04 2020
Equals (1/3)*(1 + ((1/2)*(29 + (3*sqrt(93))))^(1/3) + ((1/2)*(29 - 3*sqrt(93)))^(1/3)). See A075778. - Wolfdieter Lang, Aug 17 2022

A109134 Decimal expansion of Phi, the real root of the equation 1/x = (x-1)^2.

Original entry on oeis.org

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

Offset: 1

Lekraj Beedassy, Aug 17 2005

The silver number (A060006) is equal to Phi*(Phi-1).
Also Phi*(Phi-1) = 1/(Phi-1). - Richard R. Forberg, Oct 08 2014
Equations to which this is a root can also be written as: x = sqrt(x + sqrt(x)); x^2 - x - sqrt(x) = 0; or this form where n = 1: x = n + 1/sqrt(x). When n = 2 then the root is 2.618033988... = A104457 = 1 + A001622 or 1 + "Golden Ratio" called phi. - Richard R. Forberg, Oct 08 2014
Also equals the largest root (negated) of the Mandelbrot polynomial P_2(z) = 1+z*(1+z)^2. - Jean-François Alcover, Apr 16 2015
Suppose that r is a real number in the interval [3/2, 5/3). Let C(r) = (c(k)) be the sequence of coefficients in the Maclaurin series for 1/(Sum_{k>=0} floor((k+1)*r))(-x)^k). Conjectures: the limit L(r) of c(k+1)/c(k) as k -> oo exists, L(r) is discontinuous at 5/3 (cf. A279676), and the left limit of L(r) as r->5/3 is Phi. - Clark Kimberling, Jul 11 2017
From Wolfdieter Lang, Nov 07 2022: (Start)
This equals r + 2/3 where r is the real root of y^3 - (1/3)*y - 25/27.
The other roots of x^3 - 2*x^2 + x - 1 are (2 + w1*((25 + 3*sqrt(69))/2)^(1/3) + w2*((25 - 3*sqrt(69))/2)^(1/3))/3 = 0.1225611668... + 0.7448617668...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) and w2 = (-1 - sqrt(3)*i)/2 are the complex conjugate roots of x^3 - 1.
Using hyperbolic functions these roots are (2 - cosh((1/3)*arccosh(25/2)) + sqrt(3)*sinh((1/3)*arccosh(25/2))*i)/3, and its complex conjugate. (End)

			1.75487766624669276004950889635852869189460661777279314398928397064...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.11, p. 340.
Martin Gardner, A Gardner's Workout, pp. 124-126, A. K. Peters MA 2001.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Simon Baker, On small bases which admit countably many expansions, Journal of Number Theory, Volume 147, February 2015, Pages 515-532.
Simon Plouffe, Plouffe's Inverter .
Nikita Sidorov, Expansions in non-integer bases: Lower, middle and top orders, Journal of Number Theory, Volume 129, Issue 4, April 2009, Pages 741-754. See Prop. 2.3 p. 744.
Yuru Zou and Derong Kong, On a problem of countable expansions, Journal of Number Theory, Volume 158, January 2016, Pages 134-150. See Theorem 1.1 p. 135.
Index entries for algebraic numbers, degree 3.

Cf. A001622, A001685, A060006, A075778, A104457, A358181.

FindRoot[x^3 - 2x^2 + x - 1 == 0, {x, 1.75}, WorkingPrecision -> 128][[1, 2]] (* Robert G. Wilson v, Aug 19 2005 *)
Root[x^3-2x^2+x-1, x, 1] // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Mar 05 2013 *)

d=104;default(realprecision,d);print(k=solve(x=1,2,(x-1)^2-1/x)); for(c=0,d,z=floor(k);print1(z,",",);k=10*(k-z))

polrootsreal(x^3-2*x^2+x-1)[1] \\ Charles R Greathouse IV, Aug 15 2014

Equals 1+A075778. - R. J. Mathar, Aug 20 2008
Equals (1/6*(108+12*sqrt(69))^(1/3) + 2/(108+12*sqrt(69))^(1/3))^2. - Vaclav Kotesovec, Oct 08 2014
Equals Rho^2 where Rho is the plastic number 1.3247179572...(see A060006). - Philippe Deléham, Sep 29 2020
From Wolfdieter Lang, Nov 07 2022: (Start)
Equals (2 + ((25 + 3*sqrt(69))/2)^(1/3) + ((25 + 3*sqrt(69))/2)^(-1/3))/3.
Equals (2 + ((25 + 3*sqrt(69))/2)^(1/3) + ((25 - 3*sqrt(69))/2)^(1/3))/3.
Equals 2*(1 + cosh((1/3)*arccosh(25/2)))/3. (End)
Equals - Sum_{k>=1} Gamma(k - k/5 - 1)*Gamma(k/5 + 1)*sin(3*k*Pi/5)/(k*Pi*Gamma(k)). - Antonio Graciá Llorente, Dec 14 2024

Extended by Klaus Brockhaus and Robert G. Wilson v, Aug 19 2005

A230151 Decimal expansion of the positive real solution of the equation x^4 + x^3 - 1 = 0.

Original entry on oeis.org

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

Offset: 0

Paolo P. Lava, Oct 11 2013

Also decimal expansion of (1+(1+(1+ ... )^(1/k))^(1/k))^(1/k), with k integer and k<0. Case k=-3.

			0.8191725133961644396995711883424270403484978325537129656...

Paolo P. Lava, Table of n, a(n) for n = 0..1000 (corrected by _Sean A. Irvine_)
Index entries for algebraic numbers, degree 4.

Cf. A086106 (other real root).

Cf. A075778, A230152-A230158.

with(numtheory); P:=proc(q,h) local a,n; a:=(q+1)^(1/h);
for n from q by -1 to 1 do a:=(1+a)^(1/h);od;
print(evalf(a,1000)); end: P(1000,-3);

Root[(#^4+#^3-1)&, 2] // RealDigits[#, 10, 100]& // First (* Jean-François Alcover, Feb 18 2014 *)

polrootsreal(x^4+x^3-1)[2] \\ Charles R Greathouse IV, Feb 04 2025

A230152 Decimal expansion of the positive real solution of the equation x^(k+1)+x^k-1=0. Case k=4.

Original entry on oeis.org

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

Offset: 0

Paolo P. Lava, Oct 11 2013

Also decimal expansion of (1+(1+(1+ ... )^(1/k))^(1/k))^(1/k), with k integer and k<0. Case k=-4.

			0.8566748838545028748523248153124343698313999454937526255...

Paolo P. Lava, Table of n, a(n) for n = 0..1000 (corrected by _Sean A. Irvine_)
Index entries for algebraic numbers, degree 5

Cf. A075778, A230151, A230153-A230158, A377080.

with(numtheory); P:=proc(q,h) local a,n; a:=(q+1)^(1/h);
for n from q by -1 to 1 do a:=(1+a)^(1/h);od;
print(evalf(a,1000)); end: P(1000,-4);

Root[x^5 + x^4 - 1, 1] // RealDigits[#, 10, 100]& // First (* Jean-François Alcover, Feb 18 2014 *)

A230158 Decimal expansion of the positive real solution of the equation x^(k+1)+x^k-1=0. Case k=10.

Original entry on oeis.org

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

Offset: 0

Paolo P. Lava, Oct 11 2013

Also decimal expansion of (1+(1+(1+ ... )^(1/k))^(1/k))^(1/k), with k integer and k<0. Case k=-10.

			0.9360691110777583783971914875702962034360714782064850849...

Paolo P. Lava, Table of n, a(n) for n = 0..1000
Index entries for algebraic numbers, degree 10.

Cf. A075778, A230151-A230157.

with(numtheory); P:=proc(q,h) local a,n; a:=(q+1)^(1/h);
for n from q by -1 to 1 do a:=(1+a)^(1/h);od;
print(evalf(a,1000)); end: P(1000,-10);

Root[x^11 + x^10 - 1, 1] // RealDigits[#, 10, 100]& // First (* Jean-François Alcover, Feb 18 2014 *)

polrootsreal(x^11+x^10-1)[1] \\ Charles R Greathouse IV, Feb 07 2025

A246884 Expansion of Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * x^(4*k).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 5, 10, 17, 26, 38, 59, 101, 182, 326, 564, 945, 1566, 2622, 4476, 7750, 13455, 23231, 39837, 68101, 116611, 200526, 346137, 598438, 1034227, 1785400, 3080418, 5317009, 9187567, 15893830, 27515434, 47647774, 82513447, 142902640, 247553410, 429020710, 743846284

Offset: 0

Paul D. Hanna, Sep 06 2014

nonn

Limit a(n)/a(n+1) = t^2 = 0.569840290998053... where t = A075778 is the positive real root of 1 - x - x^5 = 0.

Diagonal of the rational function 1 / ((1-x)*(1-y) - (x*y)^5). - Seiichi Manyama, Apr 29 2025

			G.f.: A(x) = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 5*x^6 + 10*x^7 + 17*x^8 +...
where, by definition,
A(x) = 1 + x*(1 + x^4) + x^2*(1 + 2^2*x^4 + x^8)
+ x^3*(1 + 3^2*x^4 + 3^2*x^8 + x^12)
+ x^4*(1 + 4^2*x^4 + 6^2*x^8 + 4^2*x^12 + x^16)
+ x^5*(1 + 5^2*x^4 + 10^2*x^8 + 10^2*x^12 + 5^2*x^16 + x^20) +...
which is also given by the series identity:
A(x) = 1/(1-x+x^5) + 2*x^5/(1-x+x^5)^3 + 6*x^10/(1-x+x^5)^5 + 20*x^15/(1-x+x^5)^7 + 70*x^20/(1-x+x^5)^9 + 252*x^25/(1-x+x^5)^11 + 924*x^30/(1-x+x^5)^13 +...
The logarithm of the g.f. begins:
log(A(x)) = x*(1 + x^4) + x^2*(1 + 6*x^4 + x^8)/2
+ x^3*(1 + 15*x^4 + 15*x^8 + x^12)/3
+ x^4*(1 + 28*x^4 + 70*x^8 + 28*x^12 + x^16)/4
+ x^5*(1 + 45*x^4 + 210*x^8 + 210*x^12 + 45*x^16 + x^20)/5 +...
more explicitly,
log(A(x)) = x + x^2/2 + x^3/3 + x^4/4 + 6*x^5/5 + 19*x^6/6 + 36*x^7/7 + 57*x^8/8 + 82*x^9/9 + 116*x^10/10 + 199*x^11/11 + 391*x^12/12 +...
where the logarithmic derivative equals
A'(x)/A(x) = (1-x+5*x^4+6*x^5-5*x^9)/((1+x+x^2)*(1-2*x+x^2-x^3)*(1-x+2*x^3-x^5)).

Michael De Vlieger, Table of n, a(n) for n = 0..1000
Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.

Cf. A075778, A181665, A246840, A246883, A248193.

CoefficientList[Series[1/Sqrt[(1 - x + x^5)^2 - 4 x^5], {x, 0, 41}], x] (* Michael De Vlieger, Sep 10 2021 *)

/* By definition: */
{a(n)=local(A=1); A=sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2*x^(4*k)) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

/* From closed formula: */
{a(n)=local(A=1); A= 1/sqrt((1 - x - x^5)^2 - 4*x^6 +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

/* From a series identity: */
{a(n)=local(A=1+x); A=sum(m=0, n, (2*m)!/(m!)^2 * x^(5*m) / (1 - x + x^5 +x*O(x^n))^(2*m+1)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

/* From a binomial series identity: */
{a(n)=local(A=1+x); A=sum(m=0, n, x^m*(1-x^4)^(2*m+1)*sum(k=0, n, binomial(m+k, k)^2*x^(4*k)) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

/* From a binomial series identity: */
{a(n)=local(A=1+x); A=sum(m=0, n\5, x^(5*m)*sum(k=0, n-4*m, binomial(m+k, k)^2*x^k) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

/* From a binomial series identity: */
{a(n)=local(A=1+x); A=sum(m=0, n\5, x^(5*m) * sum(k=0, m, binomial(m, k)^2*x^k) / (1-x +x*O(x^n))^(2*m+1) ); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

/* From exponential formula: */
{a(n)=local(A=1); A=exp(sum(m=1, n, sum(k=0, m, binomial(2*m, 2*k)*x^(4*k)) * x^m/m) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

/* From exponential formula: */
{a(n)=local(A=1); A=exp(sum(m=1, n, ((1+x^2)^(2*m) + (1-x^2)^(2*m))/2 * x^m/m) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

/* From formula for a(n): */
{a(n)=sum(k=0, n\4, binomial(n-4*k, k)^2)}
for(n=0, 40, print1(a(n), ", "))

G.f.: Sum_{n>=0} (2*n)!/(n!)^2 * x^(5*n) / (1 - x + x^5)^(2*n+1). - Paul D. Hanna, Oct 15 2014
G.f.: Sum_{n>=0} x^n * [Sum_{k>=0} C(n+k,k)^2 * x^(4*k)] * (1-x^4)^(2*n+1).
G.f.: Sum_{n>=0} x^(5*n) * [Sum_{k>=0} C(n+k,k)^2 * x^k].
G.f.: Sum_{n>=0} x^(5*n) * [Sum_{k=0..n} C(n,k)^2 * x^k] /(1-x)^(2n+1).
G.f.: exp( Sum_{n>=1} (x^n/n) * Sum_{k=0..n} C(2*n,2*k) * x^(4*k) ).
G.f.: exp( Sum_{n>=1} (x^n/n) * ((1+x^2)^(2*n) + (1-x^2)^(2*n))/2 ).
G.f.: 1 / sqrt((1 - x - 2*x^3 - x^5)*(1 - x + 2*x^3 - x^5)).
G.f.: 1 / sqrt((1 - x - x^5)^2 - 4*x^6).
G.f.: 1 / sqrt((1 - x + x^5)^2 - 4*x^5).
a(n) = Sum_{k=0..[n/4]} C(n-4*k, k)^2.
n*a(n) = (2*n-1)*a(n-1) - (n-1)*a(n-2) + (2*n-5)*a(n-5) + 2*(n-3)*a(n-6) - (n-5)*a(n-10). - Seiichi Manyama, Aug 10 2024

A376659 Decimal expansion of a constant related to the asymptotics of A376626 and A376627.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Oct 01 2024

			3.33526020703708086029122448156335246730884987099277968...

Cf. A075778, A333198, A376621, A376626, A376627, A376658, A376660.

RealDigits[E^Sqrt[6*Log[r]^2 + 2*PolyLog[2, 1 - r^2]] /. r -> (-1 + ((25 - 3*Sqrt[69])/2)^(1/3) + ((25 + 3*Sqrt[69])/2)^(1/3))/3, 10, 105][[1]]

Equals exp(sqrt(2*(3*log(r)^2 + polylog(2, 1 - r^2)))), where r = A075778 = 0.7548776662466927600495088963585286918946... is the real root of the equation r^2*(1+r) = 1.
Equals limit_{n->infinity} A376626(n)^(1/sqrt(n)).
Equals limit_{n->infinity} A376627(n)^(1/sqrt(n)).

A210453 Decimal expansion of Sum_{n>=1} 1/(nbinomial(3n,n)).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 21 2013

			0.37121697526024703447477166607535880558762946905197...

George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press, 2006, p. 60.

Necdet Batir, On the series Sum_{k=1..oo} binomial(3k,k)^{-1} k^{-n} x^k, Proc. Indian Acad. Sci. (Math. Sci.), Vol. 115, No. 4 (2005), pp. 371-381; arXiv preprint, arXiv:math/0512310 [math.AC], 2005.
Courtney Moen, Infinite series with binomial coefficients, Math. Mag. 64 (1) (1991) 53-55.

Cf. A005809, A073010, A075778, A210462, A210463.

A075778neg := proc()
        1/3-root[3](25/2-3*sqrt(69)/2)/3 -root[3](25/2+3*sqrt(69)/2)/3;
end proc:
A210462 := proc()
        local a075778 ;
        a075778 := A075778neg() ;
        (1+1/a075778/(a075778-1))/2 ;
end proc:
A210463 := proc()
        local a075778,a210462 ;
        a075778 := A075778neg() ;
        a210462 := A210462() ;
        -1/a075778-a210462^2 ;
        sqrt(%) ;
end proc:
A210453 := proc()
        local v,x;
        v := 0.0 ;
        for x in [ A075778neg(), A210462()+I*A210463(), A210462()-I*A210463() ] do
                v := v+ x*log(1-1/x)/(3*x-2) ;
        end do:
        evalf(v) ;
end proc:
A210453() ;

RealDigits[ HypergeometricPFQ[{1, 1, 3/2}, {4/3, 5/3}, 4/27]/3, 10, 105] // First (* Jean-François Alcover, Feb 11 2013 *)

Equals Sum_{n>=1} 1/(n*A005809(n)).
Equals Integral_{x=0..1} x^2/(1-x^2+x^3) dx.
Equals Sum_(R) R*log(1-1/R)/(3*R-2) where R is summed over the set of the three constants -A075778, A210462-i*A210463 and A210462-i*A210463, i=sqrt(-1), that is, over the set of the three roots of x^3-x^2+1.
Equals (1/sqrt(23)) * (arctan(sqrt(3)/(2*phi-1)) * 18*phi/(phi^2-phi+1) - log((phi^3+1)/(phi+1)^3) * (3*sqrt(3)*phi*(1-phi))/(phi^3+1)), where phi = ((25+3*sqrt(69))/2)^(1/3) (Batir, 2005, p. 378, eq. (3.2)). - Amiram Eldar, Dec 07 2024

A210462 Decimal expansion of the real part of the complex roots of x^3-x^2+1.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 22 2013

The real root is A075778 (negated). The imaginary parts are plus or minus A210463.
Real root of 8x^3 - 8x^2 + 2x - 1: an algebraic number of degree 3. - Charles R Greathouse IV, Apr 14 2014
The denominator of this algebraic number is 2, since its double is an algebraic integer. - Charles R Greathouse IV, Nov 12 2014

			0.87743883312334638002475444817926...

Index entries for algebraic numbers, degree 3

Cf. A075778, A210463.

A075778neg := proc()
        1/3-root[3](25/2-3*sqrt(69)/2)/3 -root[3](25/2+3*sqrt(69)/2)/3;
end proc:
A210462 := proc()
        local a075778;
        a075778 := A075778neg() ;
        (1+1/a075778/(a075778-1))/2 ;
end proc:
evalf(A210462()) ;

(2^(2/3)*(25 + 3*Sqrt[69])^(1/3) + 2^(2/3)*(25 - 3*Sqrt[69])^(1/3) + 4)/12 // RealDigits[#, 10, 125]& // First (* Jean-François Alcover, Feb 20 2013 *)

real(polroots(x^3-x^2+1))[2] \\ Charles R Greathouse IV, Apr 14 2014

polrootsreal(8*x^3-8*x^2+2*x-1)[1] \\ Charles R Greathouse IV, Apr 14 2014

Equals 1/2 + 1/(2*A075778*(A075778+1)).

cp's OEIS Frontend

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