A060006 -id:A060006 - cp's OEIS frontend

A329975 Beatty sequence for 1 + x + x^2, where x is the real solution of 1/x + 1/(1+x+x^2) = 1.

4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 53, 57, 61, 65, 69, 73, 77, 81, 85, 89, 93, 97, 101, 106, 110, 114, 118, 122, 126, 130, 134, 138, 142, 146, 150, 155, 159, 163, 167, 171, 175, 179, 183, 187, 191, 195, 199, 203, 208, 212, 216, 220, 224, 228, 232

Offset: 1

Clark Kimberling, Jan 02 2020

Let x be the real solution of 1/x + 1/(1+x+x^2) = 1. Then (floor(n*x)) and (floor(n*(x^2 + x + 1))) are a pair of Beatty sequences; i.e., every positive integer is in exactly one of the sequences. See the Guide to related sequences at A329825.

Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences

Cf. A329825, A060006, A329974 (complement).

Solve[1/x + 1/(1 + x + x^2) == 1, x]
u = 1/3 (27/2 - (3 Sqrt[69])/2)^(1/3) + (1/2 (9 + Sqrt[69]))^(1/3)/3^(2/3);
u1 = N[u, 150]
RealDigits[u1, 10][[1]]  (* A060006 *)
Table[Floor[n*u], {n, 1, 50}]              (* A329974 *)
Table[Floor[n*(1 + u + u^2)], {n, 1, 50}]  (* A329975 *)
Plot[1/x + 1/(1 + x + x^2) - 1, {x, -2, 2}]

a(n) = floor(n*(1+x+x^2)), where x = 1.324717... is the constant in A060006.

A347177 Decimal expansion of real part of (i + (i + (i + (i + ...)^(1/3))^(1/3))^(1/3))^(1/3), where i is the imaginary unit.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Aug 21 2021

This is the sum of the imaginary parts of the complex roots of the cubic equation 8*r^3 + 2*r - 1 = 0 , and its real solution is A347178. - Gerry Martens, Apr 02 2024

			1.1615413999972519360879176872471740748431...

Eric Weisstein's World of Mathematics, Nested Radical

Cf. A060006, A156548, A156590, A347178.

RealDigits[(1/12) 2^(2/3) 3^(5/6) ((Sqrt[93] - 9)^(1/3) + (9 + Sqrt[93])^(1/3)), 10, 110][[1]]

(1/12)*2^(2/3)*3^(5/6)*((sqrt(93) - 9)^(1/3) + (9 + sqrt(93))^(1/3)) \\ Michel Marcus, Aug 21 2021

2*imag(polroots(8*x^3 + 2*x - 1)[3]) \\ Gerry Martens, Apr 02 2024

Equals cosh(asinh(3*sqrt(3)/2)/3). - Gerry Martens, Apr 02 2024

A018243 Inverse Euler transform of A000931.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 7, 8, 11, 13, 17, 21, 28, 34, 45, 56, 73, 92, 120, 151, 197, 250, 324, 414, 537, 687, 892, 1145, 1484, 1911, 2479, 3196, 4148, 5359, 6954, 9000, 11687, 15140, 19672, 25516, 33166, 43065, 56010, 72784, 94716, 123185, 160380, 208740, 271913, 354123, 461529, 601436, 784209, 1022505, 1333856

Offset: 1

N. J. A. Sloane, David Broadhurst

			x^3 + x^5 + x^7 + x^8 + x^9 + x^10 + 2*x^11 + 2*x^12 + 3*x^13 + 3*x^14 + ...

Joerg Arndt, Table of n, a(n) for n = 1..1000
D. J. Broadhurst and D. Kreimer, Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops, Phys. Lett. B 393, No.3-4, 403-412 (1997).
N. J. A. Sloane, Transforms

Cf. A000931, A113788.

# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(A000931):
seq(a(n), n = 1..65); # Peter Luschny, Nov 21 2022

a[n_] := (1/n)*Sum[ MoebiusMu[n/d]*Floor[ Re[ N[ RootSum[ -1-#+#^3&, #^d& ]]]] , {d, Divisors[n]}]; a[2]=0; Table[a[n], {n, 1, 65}] (* Jean-François Alcover, Oct 05 2012, after Michael Somos *)

z = PowerSeriesRing(ZZ, 'z').gen().O(30)
r = (1 - (z**2 + z**3))/(1 - z**2)
F = -z*r.derivative()/r
[sum(moebius(n//d)*F[d] for d in divisors(n))//n for n in range(1, 24)] # F. Chapoton, Apr 25 2020

a(n) = A113788(n) unless n=2. - Michael Somos, Apr 06 2012
Reciprocal of g.f. of A000931 = (1 - x^2 - x^3) / (1 - x^2) = 1 - x^3 - x^5 - x^7 - x^9 - ... = Product_{k>0} (1 - x^k)^a(n). - Michael Somos, Jul 17 2012
a(n) ~ A060006^n / n. - Vaclav Kotesovec, Oct 09 2019

More terms from Joerg Arndt, Jul 18 2012

A052954 Expansion of (2-x-x^2-x^3)/((1-x)*(1-x^2-x^3)).

Original entry on oeis.org

2, 1, 2, 2, 2, 3, 3, 4, 5, 6, 8, 10, 13, 17, 22, 29, 38, 50, 66, 87, 115, 152, 201, 266, 352, 466, 617, 817, 1082, 1433, 1898, 2514, 3330, 4411, 5843, 7740, 10253, 13582, 17992, 23834, 31573, 41825, 55406, 73397, 97230, 128802, 170626, 226031, 299427

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

For n > 2, a(n) = floor(sqrt(a(n-3)*a(n-2) + a(n-2)*a(n-1) + a(n-1)*a(n-3))). - Gerald McGarvey, Sep 19 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1025
Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1).

Cf. A000931, A023434, A060006.

a:=[2,1,2,2];; for n in [5..40] do a[n]:=a[n-1]+a[n-2]-a[n-4]; od; a; # G. C. Greubel, Oct 22 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (2-x-x^2-x^3)/((1-x)*(1-x^2-x^3)) )); // G. C. Greubel, Oct 22 2019

spec:= [S,{S=Union(Sequence(Prod(Union(Prod(Z,Z),Z),Z)), Sequence(Z))}, unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
seq(coeff(series((2-x-x^2-x^3)/((1-x)*(1-x^2-x^3)), x, n+1), x, n), n = 0 .. 40); # G. C. Greubel, Oct 22 2019

LinearRecurrence[{1,1,0,-1}, {2,1,2,2}, 40] (* G. C. Greubel, Oct 22 2019 *)

my(x='x+O('x^40)); Vec((2-x-x^2-x^3)/((1-x)*(1-x^2-x^3))) \\ G. C. Greubel, Oct 22 2019

def A052954_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((2-x-x^2-x^3)/((1-x)*(1-x^2-x^3))).list()
A052954_list(40) # G. C. Greubel, Oct 22 2019

G.f.: (2-x-x^2-x^3)/((1-x)*(1-x^2-x^3)).
a(n) = a(n-2) + a(n-3) - 1.
a(n) = 1 + Sum_{alpha=RootOf(-1+z^2+z^3)} (1/23)*(3 +7*alpha -2*alpha^2) * alpha^(-1-n).
lim n->inf a(n)/a(n-1) = positive root of 1+x-x^3 (smallest Pisot-Vijayaraghavan number, A060006) - Gerald McGarvey, Sep 19 2004
a(n) = 2*A023434(n+1) - A023434(n) - A023434(n-2) - A023434(n-3). - R. J. Mathar, Nov 28 2011
a(n) = 1 + A000931(n+3). - G. C. Greubel, Oct 22 2019

More terms from James Sellers, Jun 05 2000

A084252 A measure of how close r^n is to an integer where r is the real root of x^3-x-1, i.e.. r = (1/2 + sqrt(23/108))^(1/3) + (1/2 - sqrt(23/108))^(1/3) = 1.3247.... (Higher absolute value of a(n) means closer, negative means less than closest integer.)

Original entry on oeis.org

3, -4, 3, 13, 13, 2, 6, 2, -2, -3, 21, 5, -3, 4, -10, -18, 7, -6, 10, -139, -16, 11, -14, 39, 54, -23, 23, -39, 3479, 53, -40, 52, -158, -165, 78, -81, 148, 2429, -177, 140, -191, 657, 517, -269, 289, -563, -3923, 595, -492, 702, -2833, -1645, 933, -1041, 2156, 9021, -2012, 1740, -2590, 12872, 5304, -3242, 3756

Offset: 1

Henry Bottomley, May 22 2003

sign

			a(4)=13 since r^4 = 3.0795956..., 1/(3.0795956...-round(3.0795956...)) = 1/0.0795956... = 12.5635... and round(12.5635...) = 13.

Cf. A056581, A060006.

Positive values when n is in A051016 and negative when n is in A051017.

a(n) = round(1/(r^n - round(r^n))).

A112639 a(n) = floor(r^n) where r is the smallest Pisot number (real root r=1.3247179... of x^3-x-1).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 9, 12, 16, 22, 29, 38, 51, 67, 89, 119, 157, 209, 276, 366, 486, 643, 853, 1130, 1496, 1983, 2626, 3480, 4610, 6106, 8090, 10716, 14196, 18807, 24913, 33004, 43721, 57917, 76725, 101638, 134643, 178364, 236281, 313007, 414645

Offset: 0

Roger L. Bagula, Mar 31 2006

nonn

Andrei Vieru, Pisot Numbers and Primes, arXiv:1205.1054 [math.NT], Apr 04 2012.

Cf. A060006 (decimal expansion of r=1.32471795724475...).
Cf. A051016, A051017.
Cf. A205579 (definition using round() instead of floor()).

r = Solve[x^3 - x - 1 == 0, x][[1,1,2]]; Table[Floor[r^n], {n, 0, 50}] (* T. D. Noe, Jan 30 2012 *)

default(realprecision,110);
default(format,"g.15");
r=real(polroots(x^3-x-1)[1])
v=vector(66, n, floor(r^(n-1)) )  /* Joerg Arndt, Jan 29 2012 */

a(n) = floor(((1/2+sqrt(23/108))^(1/3) + (1/2-sqrt(23/108))^(1/3))^n). - Jwalin Bhatt, May 06 2025

Completely edited by Joerg Arndt, Jan 29 2012

A168637 a(n) = a(n-1) + a(n-2) - a(n-4) starting a(0)=0, a(1)=1, a(2)=a(3)=3.

Original entry on oeis.org

0, 1, 3, 3, 6, 8, 11, 16, 21, 29, 39, 52, 70, 93, 124, 165, 219, 291, 386, 512, 679, 900, 1193, 1581, 2095, 2776, 3678, 4873, 6456, 8553, 11331, 15011, 19886, 26344, 34899, 46232, 61245, 81133, 107479, 142380, 188614, 249861, 330996, 438477, 580859, 769475

Offset: 0

Roger L. Bagula and Gary W. Adamson, Dec 01 2009

R. Pallu de la Barriere, Optimal Control Theory, Dover Publications, New York, 1967, pages 339-344

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1).

Cf. A007307 (for a different starting vector of the Mma program).

Cf. A000931, A060006.

R:=PowerSeriesRing(Integers(), 60);
[0] cat Coefficients(R!( x*(1+2*x-x^2)/((1-x)*(1-x^2-x^3)) )); // G. C. Greubel, Apr 20 2025

LinearRecurrence[{1,1,0,-1},{0,1,3,3},50] (* or *) CoefficientList[ Series[ x*(-1-2x+x^2)/((1-x)(x^3+x^2-1)),{x,0,50}],x] (* Harvey P. Dale, Jun 22 2011 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,0,1,1]^n*[0;1;3;3])[1,1] \\ Charles R Greathouse IV, Jul 29 2016

def A168637_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1+2*x-x^2)/((1-x)*(1-x^2-x^3)) ).list()
print(A168637_list(60)) # G. C. Greubel, Apr 20 2025

Limit_{n -> oo} a(n+1)/a(n) = A060006 (also a limiting value of A000931).
G.f.: x*(1 + 2*x - x^2)/((1-x)*(1 - x^2 - x^3)). [Dec 03 2009]
a(n) = 3*A000931(n+4) + 2*A000931(n+3) - 2. [Dec 03 2009]
a(n) = a(n-2) + a(n-3) + 2. - Greg Dresden, May 18 2020

Precise definition and more formulas supplied by the Assoc. Editors of the OEIS, Dec 03 2009

A218197 Decimal expansion of the Perrin argument a (see below).

Original entry on oeis.org

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

Offset: 0

Roman Witula, Oct 23 2012

The Perrin argument a is defined by the decomposition of the known Perrin polynomial: X^3 - X - 1 = (X - t^(-1))*(X - i*sqrt(t)*e^(i*a))*(X + i*sqrt(t)*e^(-i*a)), where t = 0.754877666... (see A075778 and A060006 for the decimal expansions of t and t^(-1) respectively) is the only positive root of the polynomial x^3 + x^2 - 1 and a := arcsin(1/(2*sqrt(t^3))) (the principal value of arc sine is considered here).
The Perrin polynomial is the characteristic polynomial of the Perrin recurrence sequence (see A001608):
A(n) = A(n-2) + A(n-3), with A(0)=3, A(1)=0, and A(2)=2.
The Binet formula of this sequence has the form
A(n) = t^(-n) + i^n * t^(n/2) * (e^(i*(a + Pi)*n) + e^(-i*a*n)) = t^(-n) + 2*(-1)^n*t^(n/2)*cos((a + Pi/2)*n),
which implies the relations
A(2*n) = t^(-2*n) + 2 * (-1)^n * cos(2*a*n) * t^n, and
A(2*n-1) = t^(-2*n+1) + 2 * (-1)^(n-1) * sin((2*n-1)*a) * t^(n - 1/2).
It is proved in the paper of Witula et al. that we have
u + v + w = 0 for the respective complex values of the roots: u in (1 + t^(-1))^(1/3), v in (1 + i*sqrt(t)*e^(i*a))^(1/3) and w in (1 - i*sqrt(t)*e^(-i*a))^(1/3).

			0.8669386054934201...

R. Witula, E. Hetmaniok, and D. Slota, Sums of the powers of any order roots taken from the roots of a given polynomial, submitted to Proceedings of the 15th International Conference on Fibonacci Numbers and Their Applications, Eger, Hungary, 2012.

Cf. A075778, A060006, A001608.

ArcSin[1/(2*Root[Function[x, x^3+x^2-1], 1]^(3/2))] // RealDigits[#, 10, 120]& // First (* Jean-François Alcover, Feb 20 2014 *)

asin(1/2/real(polroots(x^3+x^2-1)[1])^1.5) \\ Charles R Greathouse IV, Dec 11 2013

Equals arccos((1-A060006)/2)/2. - Gerry Martens, Apr 16 2024

a(119) corrected by Sean A. Irvine, Apr 16 2024

A276693 a(n) = a(n-2)*a(n-3) - a(n-1); a(0) = 3, a(1) = 5, a(2) = 7.

Original entry on oeis.org

3, 5, 7, 8, 27, 29, 187, 596, 4827, 106625, 2770267, 511908608, 294867810267, 1417828655948069, 150943952469132130267, 418071880169258361764894156, 214012660834726939177944668730210267, 63105422008735225121538219609433904551328809385

Offset: 0

Christopher C. Capobianco, Sep 13 2016

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..27

int seq(int n) {int v = 3; if(n <= 2) {v = 3+2*n;} else {v = seq(n-2)*seq(n-3) - seq(n-1);} return v;}

RecurrenceTable[{a[n] ==  a[n-2]*a[n-3]-a[n-1], a[0] == 3,a[1]==5,a[2]==7}, a, {n,0, 17}]
nxt[{a_,b_,c_}]:={b,c,a*b-c}; NestList[nxt,{3,5,7},20][[All,1]] (* Harvey P. Dale, May 27 2020 *)

a(n) ~ c^(d^n), where c = 2.46982021132238000769..., d = A060006 = (1/2+sqrt(23/108))^(1/3) + (1/2-sqrt(23/108))^(1/3) = 1.32471795724474602596..., real root of the equation d*(d^2-1) = 1. - Vaclav Kotesovec, Oct 04 2016

A296140 Decimal expansion of 1/sqrt(1 + 1/sqrt(2 + 1/sqrt(3 + 1/sqrt(4 + 1/sqrt(5 + ...))))).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Dec 05 2017

			0.7837663092363964699519430776387428127041411807738774755...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Continued Fraction Constants
Eric Weisstein's World of Mathematics, Nested Radical Constant

Cf. A052119, A060006, A072449.

RealDigits[Fold[1/Sqrt[#1 + #2]&, 0, Range[100, 1, -1]], 10, 100][[1]] (* Jean-François Alcover, Dec 19 2017 *)

cp's OEIS Frontend

A329975 Beatty sequence for 1 + x + x^2, where x is the real solution of 1/x + 1/(1+x+x^2) = 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A347177 Decimal expansion of real part of (i + (i + (i + (i + ...)^(1/3))^(1/3))^(1/3))^(1/3), where i is the imaginary unit.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A018243 Inverse Euler transform of A000931.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

Extensions

A052954 Expansion of (2-x-x^2-x^3)/((1-x)*(1-x^2-x^3)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A084252 A measure of how close r^n is to an integer where r is the real root of x^3-x-1, i.e.. r = (1/2 + sqrt(23/108))^(1/3) + (1/2 - sqrt(23/108))^(1/3) = 1.3247.... (Higher absolute value of a(n) means closer, negative means less than closest integer.)

Views

Author

Keywords

Examples

Crossrefs

Formula

A112639 a(n) = floor(r^n) where r is the smallest Pisot number (real root r=1.3247179... of x^3-x-1).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A168637 a(n) = a(n-1) + a(n-2) - a(n-4) starting a(0)=0, a(1)=1, a(2)=a(3)=3.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma