A193535 -id:A193535 - cp's OEIS frontend

A057357 a(n) = floor(3*n/7).

0, 0, 0, 1, 1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 8, 8, 9, 9, 9, 10, 10, 11, 11, 12, 12, 12, 13, 13, 14, 14, 15, 15, 15, 16, 16, 17, 17, 18, 18, 18, 19, 19, 20, 20, 21, 21, 21, 22, 22, 23, 23, 24, 24, 24, 25, 25, 26, 26, 27, 27, 27, 28, 28, 29, 29, 30, 30, 30, 31, 31, 32, 32

Offset: 0

Mitch Harris

The cyclic pattern (and numerator of the gf) is computed using Euclid's algorithm for GCD.

This sequence relates to 3/7 = 0.42857142... (essentially given by A020806). It differs from the Beatty sequence A308358 for sqrt(3)/4 = 0.43301270... = A120011.

N. Dershowitz and E. M. Reingold, Calendrical Calculations, Cambridge University Press, 1997.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, NY, 1994.

G. C. Greubel, Table of n, a(n) for n = 0..5000
N. Dershowitz and E. M. Reingold, Calendrical Calculations Web Site.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).
Index to sequences related to Beatty sequences.

Floors of other ratios: A004526, A002264, A002265, A004523, A057353, A057354, A057355, A057356, A057357, A057358, A057359, A057360, A057361, A057362, A057363, A057364, A057365, A057366, A057367.

Cf. A020806, A193535, A308358.

[Floor(3*n/7): n in [0..50]]; // G. C. Greubel, Nov 02 2017

Floor[(3*Range[0,80])/7] (* Harvey P. Dale, Aug 22 2013 *)

a(n)=3*n\7 \\ Charles R Greathouse IV, Sep 02 2015

G.f.: (1+x^2+x^4)*x^3/((1-x)*(1-x^7)) - Bruce Corrigan (scentman(AT)myfamily.com), Jul 03 2002
for all m>=0 a(7m)=0 mod 3; a(7m+1)=0 mod 3; a(7m+2)= 0 mod 3; a(7m+3) = 1 mod 3; a(5m+4) = 1 mod 3; a(7m+5) = 2 mod 3; a(7m+6) = 2 mod 3 - Bruce Corrigan (scentman(AT)myfamily.com), Jul 03 2002
Sum_{n>=3} (-1)^(n+1)/a(n) = log(2)/3 (A193535). - Amiram Eldar, Sep 30 2022

A113476 Decimal expansion of (log(2) + Pi/sqrt(3))/3.

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Jan 08 2006

This number is transcendental - this follows from a result of Baker (1968) on linear forms of algebraic numbers.

			0.835648848264721053337... = A073010 + A193535.

Jolley, Summation of Series, Dover (1961), eq (79) page 16.
Murray R. Spiegel, Seymour Lipschutz, John Liu. Mathematical Handbook of Formulas and Tables, 3rd Ed. Schaum's Outline Series. New York: McGraw-Hill (2009): p. 135, equation 21.16

Ivan Panchenko, Table of n, a(n) for n = 0..1000
A. Baker, Linear forms in the logarithms of algebraic numbers (IV). Mathematika, 15 (1968) pp. 204-216
L. Euler, De fractionibus continuis observationes, The Euler Archive, Index Number 123, Section 5
Eric W. Weisstein, Euler's Series Transformation.
Index entries for transcendental numbers

Cf. A007559, A016777, A024217, A073010, A193534, A193535.

RealDigits[(Log[2]+\[Pi]/Sqrt[3])/3,10,120][[1]]  (* Harvey P. Dale, Mar 26 2011 *)

PARI
```
1/3*(log(2)+Pi/sqrt(3))
```

Equals Integral_{x = 0..1} dx/(1+x^3) = Sum_{k >= 0} (-1)^k/(3*k+1) = 1 - 1/4 + 1/7 - 1/10 + 1/13 - 1/16 + ... (see A016777). - Benoit Cloitre, Alonso del Arte, Jul 29 2011
Generalized continued fraction: 1/(1 + 1^2/(3 + 4^2/(3 + 7^2/(3 + 10^2/(3 + ... ))))) due to Euler. For a sketch proof see A024217. - Peter Bala, Feb 22 2015
Equals (1/2)*Sum_{n >= 0} n!*(3/2)^n/(Product_{k = 0..n} 3*k + 1) = (1/2)*Sum_{n >= 0} n!*(3/2)^n/A007559(n+1) (apply Euler's series transformation to Sum_{k >= 0} (-1)^k/(3*k + 1)). - Peter Bala, Dec 01 2021
From Peter Bala, Mar 03 2024: (Start)
Equals hypergeom([1/3,  1], [4/3], -1).
Gauss's continued fraction: 1/(1 + 1/(4 + 3^2/(7 + 4^2/(10 + 6^2/(13 + 7^2/(16 + 9^2/(19 + 10^2/(22 + 12^2/(25 + 13^2/(28 + ... )))))))))). (End)
Equals (1/12) * Sum_{n >= 0} (-1/2)^n * (9*n + 7)/((3*n + 2)*(n + 1)*binomial(2*n+1/3, n+1)). - Peter Bala, Mar 05 2025

A193534 Decimal expansion of (1/3) * (Pi/sqrt(3) - log(2)).

Original entry on oeis.org

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

Offset: 0

Alonso del Arte, Jul 29 2011

The formulas for this number and the constant in A113476 are exactly the same except for one small, crucial detail: the infinite sum has a denominator of 3i + 2 rather than 3i + 1, while in the closed form, log(2)/3 is subtracted from rather than added to (Pi * sqrt(3))/9.

Understandably, the typesetter for Spiegel et al. (2009) set the closed formula for this number incorrectly (as being the same as for A113476, compare equation 21.16 on the same page of that book).

			0.373550727891424180392282045394659721402855371244161773816401641964909853052219...

L. B. W. Jolley, Summation of Series, Dover, 1961, eq. (80), page 16.
J. Rivaud, Analyse, Séries, équations différentielles, Mathématiques supérieures et spéciales, Premier cycle universitaire, Vuibert, 1981, Exercice 3, p. 132.
Murray R. Spiegel, Seymour Lipschutz, John Liu. Mathematical Handbook of Formulas and Tables, 3rd Ed. Schaum's Outline Series. New York: McGraw-Hill, 2009, p. 135, equation 21.18.

Gheorghe Coserea, Table of n, a(n) for n = 0..2015
Eric W. Weisstein, Euler's Series Transformation.

Cf. A073010, A193535, A024396, A113476, A196548, A258969.

evalf((Psi(5/6)-Psi(1/3))/6, 120); # Vaclav Kotesovec, Jun 16 2015

RealDigits[(Pi Sqrt[3])/9 - (Log[2]/3), 10, 100][[1]]

(Pi/sqrt(3)-log(2))/3 \\ Charles R Greathouse IV, Jul 29 2011

default(realprecision, 98);
eval(vecextract(Vec(Str(sumalt(n=0, (-1)^(n)/(3*n+2)))), "3..-2")) \\ Gheorghe Coserea, Oct 06 2015

Equals Sum_{k >= 0} (-1)^k/(3k + 2) = 1/2 - 1/5 + 1/8 - 1/11 + 1/14 - 1/17 + ... (see A016789).
From Peter Bala, Feb 20 2015: (Start)
Equals (1/2) * Integral_{x = 0..1} 1/(1 + x^(3/2)) dx.
Generalized continued fraction: 1/(2 + 2^2/(3 + 5^2/(3 + 8^2/(3 + 11^2/(3 + ... ))))) due to Euler. For a sketch proof see A024396. (End)
Equals (Psi(5/6)-Psi(1/3))/6. - Vaclav Kotesovec, Jun 16 2015
Equals Integral_{x = 1..infinity} 1/(1 + x^3) dx. - Robert FERREOL, Dec 23 2016
Equals (1/2)*Sum_{n >= 0} n!*(3/2)^n/(Product_{k = 0..n} 3*k + 2) = (1/2)*Sum_{n >= 0} n!*(3/2)^n/A008544(n+1) (apply Euler's series transformation to Sum_{k >= 0} (-1)^k/(3*k + 2)). - Peter Bala, Dec 01 2021
From Bernard Schott, Jan 28 2022: (Start)
Equals Integral_{x = 0..1} x/(1+ x^3) dx (see Rivaud reference).
Equals 3 * A196548. (End)
From Peter Bala, Mar 03 2024: (Start)
Equals (1/2)*hypergeom([2/3, 1], [5/3], -1).
Gauss's continued fraction: 1/(2 + 2^2/(5 + 3^2/(8 + 5^2/(11 + 6^2/(14 + 8^2/(17 + 9^2/(20 + 11^2/(23 + 12^2/(26 + ... ))))))))). (End)

A372609 Decimal expansion of (10/9)*log(2).

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, May 07 2024

			0.77016353395549478824136902384241840897277792706695028235631...

Paolo Xausa, Table of n, a(n) for n = 0..10000
J. M. Borwein and P. B. Borwein, Strange Series and High Precision Fraud, The American Mathematical Monthly, Vol. 99, No. 7 (1992), pp. 622-640.
Index entries for transcendental numbers

Cf. A002162, A193535, A196564, A372585.

Mathematica
```
First[RealDigits[10*Log[2]/9, 10, 100]]
```

log(2)*10/9 \\ Charles R Greathouse IV, May 18 2024

Equals Sum_{k >= 1} A196564(k)/(k*(k + 1)). See eq. 2.6 and Sum 13 in Borwein and Borwein (1992), p. 626.

Equals (10/3)*A193535. - Hugo Pfoertner, May 07 2024

A387235 Decimal expansion of 2*log(2)/3.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Aug 23 2025

Area enclosed by the curve of the equation x^6 + y^6 - x^3*y + x*y^3 = 0.

The asymptotic mean of A256232. - Amiram Eldar, Aug 23 2025

			0.46209812037329687294482141430545104538366675624...

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 436.
Index entries for transcendental numbers.

Cf. A002162, A001504, A010701, A010722, A016627, A193535, A256232.

Mathematica
```
RealDigits[2Log[2]/3,10,100][[1]]
```

Equals A002162*A010722.
Equals log(4)/3 = A010701*A016627.
Equals Sum_{k>=0} (-1)^k/((3*k + 1)*(3*k + 2)) = Integral_{x=0..1} x^2*log(1 + 1/x^3) = -Integral_{x=0..1} log[1 - x^6]/x^4. [Shamos]
Equals A016627/3 = 2*A193535. - Hugo Pfoertner, Aug 23 2025

A375370 Decimal expansion of log(2)/3 + zeta(3)/(2*Pi^2).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Aug 13 2024

			0.29194597430343497803291376809503507598583434521003...

Olivier Espinosa and Victor H. Moll, On some integrals involving the Hurwitz zeta function: Part 1, Raman. J. 6 (2002) 159-188, eq. (5.7).

Cf. A193535.

Maple
```
log(2)/3+Zeta(3)/2/Pi^2 ; evalf(%) ;
```

RealDigits[Log[2] / 3 + Zeta[3] / (2*Pi^2), 10, 120][[1]] (* Amiram Eldar, Aug 19 2024 *)

Equals -Integral_{x=0..1} x^2* log(sin(Pi*x)) dx.

cp's OEIS Frontend

A057357 a(n) = floor(3*n/7).

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A113476 Decimal expansion of (log(2) + Pi/sqrt(3))/3.

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A193534 Decimal expansion of (1/3) * (Pi/sqrt(3) - log(2)).

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A372609 Decimal expansion of (10/9)*log(2).

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A387235 Decimal expansion of 2*log(2)/3.

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A375370 Decimal expansion of log(2)/3 + zeta(3)/(2*Pi^2).

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