A088559 -id:A088559 - cp's OEIS frontend

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

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

A376621 Decimal expansion of a constant related to the asymptotics of A369557 and A376580.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Sep 30 2024

			2.75108509088891993943420496204894703641817860263717...

Cf. A333198, A369557, A376542, A376580, A376623, A376658, A376659, A376660.

Cf. A088559.

RealDigits[E^Sqrt[3*Log[r]^2/2 + 4*PolyLog[2, r^(1/2)] - Pi^2/3] /. r -> (-2 + ((29 - 3*Sqrt[93])/2)^(1/3) + ((29 + 3*Sqrt[93])/2)^(1/3))/3, 10, 120][[1]] (* Vaclav Kotesovec, Oct 07 2024 *)

Equals limit_{n->infinity} A369557(n)^(1/sqrt(n)).
Equals limit_{n->infinity} A376580(n)^(1/sqrt(n)).
Equals limit_{n->infinity} A376542(n)^(1/sqrt(n)).
Equals limit_{n->infinity} A376623(n)^(1/sqrt(n)).
Equals exp(sqrt(3*log(r)^2/2 + 4*polylog(2, r^(1/2)) - Pi^2/3)), where r = A088559 = 0.465571231876768026656731... is the real root of the equation r*(1+r)^2 = 1. - Vaclav Kotesovec, Oct 07 2024

A246840 Expansion of Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * x^(2*k).

Original entry on oeis.org

1, 1, 1, 2, 5, 10, 18, 35, 73, 151, 306, 623, 1286, 2668, 5531, 11477, 23889, 49852, 104175, 217936, 456534, 957609, 2010839, 4226417, 8891022, 18719637, 39443860, 83170162, 175484915, 370491775, 782648333, 1654197568, 3498049053, 7400639286, 15664103420, 33168342557, 70260909811

Offset: 0

Paul D. Hanna, Sep 04 2014

nonn

Compare to the g.f. of Narayana's cows sequence A000930:
Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k) * x^(2*k)  =  1/(1-x-x^3).
Compare to the g.f. of Whitney numbers sequence A051286:
Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * x^k = 1/sqrt((1+x+x^2)*(1-3*x+x^2)).
...
Lim_{n->infinity} a(n)/a(n+1) = t^2 = 0.465571231876768... (A088559) where t = ((sqrt(93)+9)/18)^(1/3) - ((sqrt(93)-9)/18)^(1/3) is the positive real root of 1 - x - x^3 = 0.
Diagonal of the rational function 1 / ((1 - x)*(1 - y) - (x*y)^3). - Ilya Gutkovskiy, Apr 23 2025

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 10*x^5 + 18*x^6 + 35*x^7 + ...
where, by definition,
A(x) = 1 + x*(1 + x^2) + x^2*(1 + 2^2*x^2 + x^4)
+ x^3*(1 + 3^2*x^2 + 3^2*x^4 + x^6)
+ x^4*(1 + 4^2*x^2 + 6^2*x^4 + 4^2*x^6 + x^8)
+ x^5*(1 + 5^2*x^2 + 10^2*x^4 + 10^2*x^6 + 5^2*x^8 + x^10) + ...
which is also given by the series identity:
A(x) = 1/(1-x+x^3) + 2*x^3/(1-x+x^3)^3 + 6*x^6/(1-x+x^3)^5 + 20*x^9/(1-x+x^3)^7 + 70*x^12/(1-x+x^3)^9 + 252*x^15/(1-x+x^3)^11 + 924*x^18/(1-x+x^3)^13 + ...
The logarithm of the g.f. begins:
log(A(x)) = x*(1 + x^2) + x^2*(1 + 6*x^2 + x^4)/2
+ x^3*(1 + 15*x^2 + 15*x^4 + x^6)/3
+ x^4*(1 + 28*x^2 + 70*x^4 + 28*x^6 + x^8)/4
+ x^5*(1 + 45*x^2 + 210*x^4 + 210*x^6 + 45*x^8 + x^10)/5 + ...
more explicitly,
log(A(x)) = x + x^2/2 + 4*x^3/3 + 13*x^4/4 + 26*x^5/5 + 46*x^6/6 + 99*x^7/7 + 229*x^8/8 + 499*x^9/9 + 1046*x^10/10 + 2223*x^11/11 + 4810*x^12/12 + ...
where the logarithmic derivative equals
A'(x)/A(x) = (1-x+3*x^2+4*x^3-3*x^5)/((1-x+2*x^2-x^3)*(1-x-2*x^2-x^3)).

G. C. Greubel, 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. A088559, A246861, A181665, A246883, A246884, A248193.

CoefficientList[Series[1/Sqrt[(1 - x - x^3)^2 - 4*x^4], {x,0,50}], x] (* G. C. Greubel, Apr 27 2017 *)

/* By definition: */
{a(n)=local(A=1);A=sum(m=0,n,x^m*sum(k=0,m,binomial(m,k)^2*x^(2*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^3)^2 - 4*x^4 +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^(3*m) / (1 - x + x^3 +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^2)^(2*m+1)*sum(k=0, n, binomial(m+k, k)^2*x^(2*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\3, x^(3*m)*sum(k=0, n-3*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\3, x^(3*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^(2*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*m) + (1-x)^(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\2,binomial(n-2*k,k)^2)}
for(n=0,40,print1(a(n),", "))

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

A376660 Decimal expansion of a constant related to the asymptotics of A376630 and A376631.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Oct 01 2024

			2.045390691852050639893704244342601252265948793467833187994662870934455617...

Cf. A333198, A376621, A376630, A376631, A376658, A376659, A376815.

Cf. A088559.

RealDigits[E^Sqrt[3*Log[r]^2/4 + 2*PolyLog[2, r^(1/2)] - Pi^2/6] /. r -> (-2 + ((29 - 3*Sqrt[93])/2)^(1/3) + ((29 + 3*Sqrt[93])/2)^(1/3))/3, 10, 120][[1]] (* Vaclav Kotesovec, Oct 07 2024 *)

Equals limit_{n->infinity} A376630(n)^(1/sqrt(n)).
Equals limit_{n->infinity} A376631(n)^(1/sqrt(n)).
Equals A376815^(1/2). - Vaclav Kotesovec, Oct 06 2024
Equals exp(sqrt(3*log(r)^2/4 + 2*polylog(2, r^(1/2)) - Pi^2/6)), where r = A088559 = 0.4655712318767680266567312252199... is the real root of the equation r*(1+r)^2 = 1. - Vaclav Kotesovec, Oct 07 2024

A376815 Decimal expansion of a constant related to the asymptotics of A376812.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Oct 05 2024

			4.18362308231501037592434207471436289895638697707035887857832710...

Cf. A333198, A376621, A376658, A376659, A376660, A376812.

Cf. A088559.

RealDigits[E^Sqrt[3*Log[r]^2 + 8*PolyLog[2, r^(1/2)] - 2*Pi^2/3] /. r -> (-2 + ((29 - 3*Sqrt[93])/2)^(1/3) + ((29 + 3*Sqrt[93])/2)^(1/3))/3, 10, 120][[1]] (* Vaclav Kotesovec, Oct 07 2024 *)

Equals limit_{n->infinity} A376812(n)^(1/sqrt(n)).
Equals A376660^2. - Vaclav Kotesovec, Oct 06 2024
Equals exp(sqrt(3*log(r)^2 + 8*polylog(2, r^(1/2)) - 2*Pi^2/3)), where r = A088559 = 0.4655712318767680266567312252199... is the real root of the equation r*(1+r)^2 = 1. - Vaclav Kotesovec, Oct 07 2024

A358181 Decimal expansion of the real root of x^3 - 2*x^2 - x - 1.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Nov 07 2022

This equals r0 + 2/3 where r0 is the real root of y^3 - (7/3)*y - 61/27.
The other roots of x^3 - 2*x^2 - x - 1 are (2 + w1*((61 + 9*sqrt(29))/2)^(1/3) + w2*((61 - 9*sqrt(29))/2)^(1/3))/3 = -0.2734091384... + 0.5638210928...*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 - sqrt(7)*(cosh((1/3)*arccosh((61/98)*sqrt(7))) - sqrt(3)*sinh((1/3)*arccosh((61/98)*sqrt(7)))*i))/3, and its complex conjugate.

			2.5468182768840820791359975088097915288112703374520061295514745747111979831...

Cf. A088559, A231187, 1 - A160389, A255240.

RealDigits[x /. FindRoot[x^3 - 2*x^2 - x - 1, {x, 2}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Nov 08 2022 *)
RealDigits[Root[x^3-2x^2-x-1,1],10,120][[1]] (* Harvey P. Dale, Mar 30 2025 *)

r = (2 + ((61 + 9*sqrt(29))/2)^(1/3) + 7*((61 + 9*sqrt(29))/2)^(-1/3))/3.
r = (2 + ((61 + 9*sqrt(29))/2)^(1/3) + ((61 - 9*sqrt(29))/2)^(1/3))/3.
r = 2*(1 + sqrt(7)*cosh((1/3)*arccosh((61/98)*sqrt(7))))/3.
r = (2/3) +(2^(2/3)*61^(1/3))/3*Hyper2F1([-1/6,1/3],[1/2],2349/3721). - Gerry Martens, Nov 08 2022

A144229 The numerators of the convergents to the recursion x=1/(x^2+1).

Original entry on oeis.org

1, 1, 4, 25, 1681, 5317636, 66314914699609, 8947678119828215014722891025, 178098260698995011212395018312912894502905113202338936836

Offset: 0

Cino Hilliard, Sep 15 2008

The recursion converges to the real root of 1/(x^2+1) - x = 0, 0.682327803...
An interesting consequence of this result occurs if we multiply by x^2+1 to get 1-x-x^3=0. These different equations intersect at the same root 0.682327803... Note also that a(n) is a square. The square roots form sequence A076725.
a(n) is the number of (0,1)-labeled perfect binary trees of height n such that no adjacent nodes have 1 as the label and the root is labeled 0. - Ran Pan, May 22 2015

Ran Pan, Exercise R, Project P.

Cf. A076725, A088559.

f[n_]:=(n+1/n)/n;Prepend[Denominator[NestList[f,2,7]],1] (* Vladimir Joseph Stephan Orlovsky, Nov 19 2010 *)
RecurrenceTable[{a[n]==(a[n-2]^2 + a[n-1])^2, a[0]==1, a[1]==1},a,{n,0,10}] (* Vaclav Kotesovec, May 22 2015 after Ran Pan *)

x=0;for(j=1,10,x=1/(x^2+1);print1((numerator(x))","))

a(n+2) = (a(n)^2 + a(n+1))^2. - Ran Pan, May 22 2015

a(n) ~ c * d^(2^n), where c = A088559 = 0.465571231876768... is the root of the equation c*(1+c)^2 = 1, d = 1.6634583970724267140029... . - Vaclav Kotesovec, May 22 2015

cp's OEIS Frontend

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

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A376621 Decimal expansion of a constant related to the asymptotics of A369557 and A376580.

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A246840 Expansion of Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * x^(2*k).

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A376660 Decimal expansion of a constant related to the asymptotics of A376630 and A376631.

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A376815 Decimal expansion of a constant related to the asymptotics of A376812.

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A358181 Decimal expansion of the real root of x^3 - 2*x^2 - x - 1.

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A144229 The numerators of the convergents to the recursion x=1/(x^2+1).

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