cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A020759 Decimal expansion of (-1)*Gamma'(1/2)/Gamma(1/2) where Gamma(x) denotes the Gamma function.

Original entry on oeis.org

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

Views

Author

Benoit Cloitre, May 24 2003

Keywords

Comments

Decimal expansion of -psi(1/2). - Benoit Cloitre, Mar 07 2004

Examples

			1.96351002602142347944097633299875556719315960466...
		

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), 6.3.3, p. 258. - Robert G. Wilson v, Jun 20 2011
  • S. J. Patterson, An introduction to the theory of the Riemann zeta function, Cambridge studies in advanced mathematics no. 14, p. 135.

Crossrefs

Programs

  • Magma
    R:=RealField(100); EulerGamma(R) + 2*Log(2); // G. C. Greubel, Aug 27 2018
  • Maple
    evalf(-Psi(0.5)) ; # R. J. Mathar, Sep 10 2013
  • Mathematica
    RealDigits[ EulerGamma + 2 Log[2], 10, 111][[1]] (* Robert G. Wilson v, Jun 20 2011 *)
  • PARI
    Euler+2*log(2)
    
  • PARI
    2-psi(-1/2) \\ Stanislav Sykora, Oct 03 2014
    

Formula

Gamma'(1/2)/Gamma(1/2) = -EulerGamma - 2*log(2) = -1.9635100260214234794... where EulerGamma is the Euler-Mascheroni constant (A001620).
Equals 2 - psi(-1/2) = 2-A248176. - Stanislav Sykora, Oct 03 2014
Equals A131265/A002161. - R. J. Mathar, Jun 02 2022
Equals lim_{n->oo} (Sum_{k=0..n} 1/(k+1/2) - log(n)). - Amiram Eldar, Mar 04 2023

A155172 Decimal expansion of log_2 (20).

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Oct 30 2009

Keywords

Comments

Equals 2 + A020858 = 1 + A020862 = A016643 / A002162. - Michel Marcus, Jul 28 2013

Examples

			4.3219280948873623478703194294893901758648313930245806120547...
		

Crossrefs

Cf. decimal expansion of log_2(m): A020857 (m=3), A020858 (m=5), A020859 (m=6), A020860 (m=7), A020861 (m=9), A020862 (m=10), A020863 (m=11), A020864 (m=12), A152590 (m=13), A154462 (m=14), A154540 (m=15), A154847 (m=17), A154905 (m=18), A154995 (m=19), this sequence, A155536 (m=21), A155693 (m=22), A155793 (m=23), A155921 (m=24).

Programs

  • Mathematica
    RealDigits[Log[2, 20], 10, 100][[1]] (* Vincenzo Librandi, Aug 29 2013 *)

A307704 Expansion of (1/(1 - x)) * Sum_{k>=1} (-x)^k/(1 - (-x)^k).

Original entry on oeis.org

-1, 1, -1, 2, 0, 4, 2, 6, 3, 7, 5, 11, 9, 13, 9, 14, 12, 18, 16, 22, 18, 22, 20, 28, 25, 29, 25, 31, 29, 37, 35, 41, 37, 41, 37, 46, 44, 48, 44, 52, 50, 58, 56, 62, 56, 60, 58, 68, 65, 71, 67, 73, 71, 79, 75, 83, 79, 83, 81, 93, 91, 95, 89, 96, 92, 100, 98, 104, 100, 108
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 22 2019

Keywords

Crossrefs

Cf. A001620 (gamma), A002162.

Programs

  • Mathematica
    nmax = 70; Rest[CoefficientList[Series[1/(1 - x) Sum[(-x)^k/(1 - (-x)^k), {k, 1, nmax}], {x, 0, nmax}], x]]
    Table[Sum[(-1)^k DivisorSigma[0, k], {k, 1, n}], {n, 1, 70}]
    Accumulate[Array[(-1)^#*DivisorSigma[0, #] &, 70]] (* Amiram Eldar, Oct 14 2022 *)

Formula

a(n) = Sum_{k=1..n} (-1)^k*A000005(k).
a(n) = n*log(n)/2 + (gamma - log(2) - 1/2)*n + O(n^(131/416 + eps)) (Tóth, 2017). - Amiram Eldar, Oct 14 2022

A016631 Decimal expansion of log(8).

Original entry on oeis.org

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

Views

Author

Keywords

Comments

a(n+1) is also the sequence of digits in the base-ten expansion of the number representing the probability that an acute triangle could be formed with the pieces obtained by breaking a stick into three parts at random. The breaking points are chosen with uniform distribution and independently of one another. - Eugen J. Ionascu, Feb 19 2011

Examples

			2.079441541679835928251696364374529704226500403080765762362040028480180....
		

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.
  • Bruce C. Berndt, Ramanujan's Notebooks Part I, Springer-Verlag.

Crossrefs

Cf. A016736 (continued fraction). - Harry J. Smith, May 16 2009

Programs

  • Maple
    a:=proc(n)
      local x,y,z,w;
        Digits:=2*n+1;
         x:=3*ln(2);y:=floor(10^(n-2)*x)*10;
           z:=floor(10^(n-1)*x);w:=z-y;
    end: # Eugen J. Ionascu, Feb 19 2011
  • Mathematica
    RealDigits[Log[8], 10, 90][[1]] (* Bruno Berselli, Mar 26 2013 *)
  • PARI
    default(realprecision, 20080); x=log(8); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016631.txt", n, " ", d)); \\ Harry J. Smith, May 16 2009

Formula

Equals 2 + Sum_{n >= 1} 1/( n*(16*n^2 - 1) ). This summation was the first problem submitted by Ramanujan to the Journal of the Indian Mathematical Society. See Berndt, Corollary on p. 29. - Peter Bala, Feb 25 2015
Equals 2 + Sum_{n >= 1} (-1)^n*(n-1)/(n*(n+1)). - Bruno Berselli, Sep 09 2020
Equals 2 + Sum_{k>=1} zeta(2*k+1)/16^k. - Amiram Eldar, May 27 2021
Equals 3*A002162. - R. J. Mathar, Apr 11 2024

A253191 Decimal expansion of log(2)^2.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Mar 24 2015

Keywords

Examples

			0.480453013918201424667102526326664971730552951594545586866864...
		

Crossrefs

Programs

Formula

Integral_{0..1} log(1-x^2)/(x*(1+x)) dx = -log(2)^2.
Integral_{0..1} log(log(1/x))/(x+sqrt(x)) dx = log(2)^2.
Equals Sum_{k>=1} H(k)/(2^k * (k+1)) = 2 * Sum_{k>=1} (-1)^(k+1) * H(k)/(k+1), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, Aug 05 2020
Equals Sum_{n >= 0} (-1)^n/(2^(n+1)*(n+1)^2*binomial(2*n+1,n)). See my entry in A002544 dated Apr 18 2017. Cf. A091476. - Peter Bala, Jan 30 2023
Equals 2*Integral_{x=-1..1} (abs(x)*log(x^2 + 1))/(x^2 + 1) dx. - Kritsada Moomuang, May 27 2025

A267315 Decimal expansion of the Dirichlet eta function at 4.

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Jan 13 2016

Keywords

Examples

			eta(4) = 1/1^4 - 1/2^4 + 1/3^4 - 1/4^4 + 1/5^4 - 1/6^4 + ... = 0.9470328294972459175765032344735219149279070829288860...
		

Crossrefs

Programs

  • Magma
    pi:= 7*Pi(RealField(110))^4 / 720; Reverse(Intseq(Floor(10^100*pi))); // Vincenzo Librandi, Feb 04 2016
    
  • Mathematica
    RealDigits[(7 Pi^4)/720, 10, 120][[1]]
  • PARI
    7*Pi^4/720 \\ Michel Marcus, Feb 01 2016
    
  • Sage
    s = RLF(0); s
    RealField(110)(s)
    for i in range(1,10000): s += -((-1)^i/((i)^4))
    print(s) # Terry D. Grant, Aug 04 2016

Formula

eta(4) = Sum_{k > 0} (-1)^(k+1)/k^4 = (7*Pi^4)/720.
eta(4) = Lim_{n -> infinity} A120296(n)/A334585(n) = (7/8)*A013662. - Petros Hadjicostas, May 07 2020

A016629 Decimal expansion of log(6).

Original entry on oeis.org

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

Views

Author

Keywords

Examples

			1.791759469228055000812477358380702272722990692183004705855374343130887...
		

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.

Crossrefs

Cf. A016734 (continued fraction).

Programs

  • Mathematica
    First[RealDigits[Log[6], 10, 100]] (* Paolo Xausa, Mar 21 2024 *)
  • PARI
    default(realprecision, 20080); x=log(6); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016629.txt", n, " ", d)); \\ Harry J. Smith, May 16 2009

Formula

log(6) = 2*Sum_{n >= 1} 1/(n*P(n, 7/5)*P(n-1, 7/5)), where P(n, x) denotes the n-th Legendre polynomial. The first 20 terms of the series gives the approximation log(6) = 1.79175946922805(27...), correct to 14 decimal places. - Peter Bala, Mar 19 2024
Equals A002162 + A002391. - R. J. Mathar, Jun 10 2024

A004443 Nimsum n + 2.

Original entry on oeis.org

2, 3, 0, 1, 6, 7, 4, 5, 10, 11, 8, 9, 14, 15, 12, 13, 18, 19, 16, 17, 22, 23, 20, 21, 26, 27, 24, 25, 30, 31, 28, 29, 34, 35, 32, 33, 38, 39, 36, 37, 42, 43, 40, 41, 46, 47, 44, 45, 50, 51, 48, 49, 54, 55, 52, 53, 58, 59, 56, 57, 62, 63, 60, 61, 66, 67, 64, 65
Offset: 0

Views

Author

Keywords

Comments

A self-inverse permutation of the natural numbers. - Philippe Deléham, Nov 22 2016

References

  • E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 60.
  • J. H. Conway, On Numbers and Games. Academic Press, NY, 1976, pp. 51-53.

Crossrefs

Essentially the same as A256008 - 1.
Also the second column of A274528.
Cf. A002162.

Programs

  • Maple
    nimsum := proc(a,b) local t1,t2,t3,t4,l; t1 := convert(a+2^200,base,2); t2 := convert(b+2^200,base,2); t3 := evalm(t1+t2); map(x->x mod 2, t3); t4 := convert(evalm(%),list); l := convert(t4,base,2,10); sum(l[k]*10^(k-1), k=1..nops(l)); end;
    f := n -> n + 2*(-1)^floor(n/2); # N. J. A. Sloane, Jul 06 2019
  • Mathematica
    Table[BitXor[n, 2], {n, 0, 100}] (* T. D. Noe, Feb 09 2013 *)
  • PARI
    a(n)=bitxor(n,2) \\ Charles R Greathouse IV, Oct 07 2015
    
  • Python
    for n in range(20): print(2^n) # Oliver Knill, Feb 16 2020

Formula

a(n) = n XOR 2. - Joerg Arndt, Feb 07 2013
G.f.: (2-x-2x^2+3x^3)/((1-x)^2(1+x^2)). - Ralf Stephan, Apr 24 2004
The sequences 'Nimsum n + m' seem to have the general o.g.f. p(x)/q(x) with p, q polynomials and q(x) = (1-x)^2*Product_{k>=0} (1+x^(2^e(k))), with Sum_{k>=0} 2^e(k) = m. - Ralf Stephan, Apr 24 2004
a(n) = n + 2(-1)^floor(n/2). - Mitchell Harris, Jan 10 2005
a(n) = OR(n,2) - AND(n,2). - Gary Detlefs, Feb 06 2013
E.g.f.: 2*(sin(x) + cos(x)) + x*exp(x). - Ilya Gutkovskiy, Jul 01 2016
Sum_{n>=0,n<>2} (-1)^n/a(n) = -log(2) = -A002162. - Peter McNair, Aug 07 2023

A100199 Decimal expansion of Pi^2/(12*log(2)), inverse of Levy's constant.

Original entry on oeis.org

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

Views

Author

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Dec 27 2004

Keywords

Comments

From A.H.M. Smeets, Jun 12 2018: (Start)
The denominator of the k-th convergent obtained from a continued fraction of a constant, the terms of the continued fraction satisfying the Gauss-Kuzmin distribution, will tend to exp(k*A100199).
Similarly, the error between the k-th convergent obtained from a continued fraction of a constant, and the constant itself will tend to exp(-2*k*A100199). (End)
The term "Lévy's constant" is sometimes used to refer to this constant (Wikipedia). - Bernard Schott, Sep 01 2022

Examples

			1.1865691104156254528217229759472371205683565364720543359542542986528...
		

References

  • Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.7, p. 54.

Crossrefs

Programs

Formula

Equals 1/A089729 = log(A086702) = A174606/2.
Equals ((Pi^2)/12)/log(2) = A072691 / A002162 = (Sum_{n>=1} ((-1)^(n+1))/n^2) / (Sum_{n>=1} ((-1)^(n+1))/n^1). - Terry D. Grant, Aug 03 2016
Equals (-1/log(2)) * Integral_{x=0..1} log(x)/(1+x) dx (from Corless, 1992). - Bernard Schott, Sep 01 2022

A132270 a(n) = floor((n^7-1)/(7*n^6)), which is the same as integers repeated 7 times.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10
Offset: 1

Views

Author

Mohammad K. Azarian, Nov 06 2007

Keywords

Crossrefs

Cf. A004526 ([n/2]), A002264 ([n/3]), A002265 ([n/4]), A002266 ([n/5]), A054895.
Cf. A152467 ([n/6]), A132292 ([(n-1)/8]).
Cf. A002162.

Programs

Formula

a(n) = floor((n^7-n^6)/(7*n^6-6*n^5)). - Mohammad K. Azarian, Nov 08 2007
G.f.: x^8/(1-x-x^7+x^8). - Robert Israel, Feb 02 2015
a(n) = a(n-1)+a(n-7)-a(n-8). - Wesley Ivan Hurt, May 03 2021
a(n) = floor((n-1)/7). - M. F. Hasler, May 19 2021
Sum_{n>=8} (-1)^n/a(n) = log(2) (A002162). - Amiram Eldar, Sep 30 2022

Extensions

Offset corrected by Mohammad K. Azarian, Nov 19 2008
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