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.

Showing 1-4 of 4 results.

A007525 Decimal expansion of log_2 e.

Original entry on oeis.org

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

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Author

N. J. A. Sloane

Keywords

Comments

Around 1670, James Gregory discovered by inversion of 1 - 1/2 + 1/3 - 1/4 + 1/5 - ... = log(2) that 1 + 1/2 - 1/12 + 1/24 - 19/720 + (27/1440 = 3/160) - 863/60480 + ... = 1/log(2). See formula with A002206 and A002207. See also A141417 signed /A091137; case i = 0 in A165313. First row in array p. 36 of the reference. - Paul Curtz, Sep 12 2011
This constant 1/log(2) is also related to the asymptotic evaluation of the maximum number of subtraction steps required to compute gcd(m, n) by the binary Euclidean algorithm, m and n being odd and chosen at random. - Jean-François Alcover, Jun 23 2014, after Steven Finch

Examples

			1.442695040888963407359924681...

References

  • Paul Curtz, Intégration numérique des systèmes différentiels .. , note n° 12, Centre de Calcul Scientifique de l'Armement, Arcueil, 1969.
  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.18 Porter-Hensley constants, p. 159.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 25, equation 25:14:3 at page 232.

Links

Crossrefs

Cf. A000670, A002206, A002207, A052882, A091137, A141417, A165313.

Programs

Formula

Equals lim_{n->infinity} A000670(n)/A052882(n). - Mats Granvik, Aug 10 2009
Equals Sum_{k>=-1} A002206(k)/A002207(k). - Paul Curtz, Sep 12 2011
Also equals integral_{x>=2} 1/(x*log(x)^2). - Jean-François Alcover, May 24 2013
1/log(2) = Sum_{n = -infinity..infinity} (2^n / (1 + 2^2^n)). - Nicolas Nagel, Mar 16 2018
More generally: 1/log(2) = Sum_{n = -infinity..infinity} (2^(n+x) / (1 + 2^2^(n+x))) for all real x. - Nicolas Nagel, Jul 02 2019
From Amiram Eldar, Jun 04 2023: (Start)
Equals 1 + Sum_{k>=1} 1/(2^k * (1 + 2^(1/2^k))).
Equals Product_{k>=1} ((1 + 2^(1/2^k))/2). (End)

A174727 a(n) = A091137(n+1)/(n+1).

Original entry on oeis.org

2, 6, 8, 180, 288, 10080, 17280, 453600, 806400, 47900160, 87091200, 217945728000, 402361344000, 2241727488000, 4184557977600, 2000741783040000, 3766102179840000, 2838385676206080000, 5377993912811520000, 1686001091666411520000, 3211430650793164800000, 423033001181754163200000
Offset: 0

Views

Author

Paul Curtz, Mar 28 2010

Keywords

Comments

Previous name: Inverse Akiyama-Tanigawa algorithm. From a column instead of a row. Bernoulli case A164555/A027642. We start from column 1, 1/2, 1/3, 1/4, 1/5 = A000012/A000027. First row: 1) (unreduced) 1, 1/2, 5/12, 9/24, 251/720 = A002657/A091137 (Cauchy from Bernoulli) (*); 2) (reduced) 1, 1/2, 5/12, 3/8, 251/720 = A002208/A002209 (Stirling and Bernoulli). Unreduced second row: 1/2, 1/6, 1/8, 19/180, 27/288, 863/10080 = A141417(n+1)/a(n).
(*) Reference page 56 (first row) and page 36 (upper main diagonal). From J. C. Adams (and Bashforth) numerical integration. See A165313 and A147998. See A002206 logarithm numbers (Gregory).

References

  • P. Curtz, Intégration numérique des systèmes différentiels .. . Note 12, Centre de Calcul Scientifique de l'Armement, Arcueil, 1969.

Crossrefs

Cf. A002689, A091137.

Programs

  • Mathematica
    A091137[n_] := A091137[n] = Product[d, {d, Select[ Divisors[n] + 1, PrimeQ]}]*A091137[n-1]; A091137[0] = 1; a[n_] := A091137[n+1]/(n+1); Table[a[n], {n, 0, 18}] (* Jean-François Alcover_, Aug 14 2012 *)
  • PARI
    f(n) = my(r =1); forprime(p=2, n+1, r*=p^(n\(p-1))); r; \\ A091137
a(n) = f(n+1)/(n+1); \\ Michel Marcus, Jun 30 2019

Formula

a(n) = A091137(n+1)/(n+1).

Extensions

Extended up to a(18) by Jean-François Alcover, Aug 14 2012
New name and more terms from Michel Marcus, Jun 30 2019

A232853 Repeat n+1 times A091137(n).

Original entry on oeis.org

1, 2, 2, 12, 12, 12, 24, 24, 24, 24, 720, 720, 720, 720, 720, 1440, 1440, 1440, 1440, 1440, 1440, 60480, 60480, 60480, 60480, 60480, 60480, 60480, 120960, 120960, 120960, 120960, 120960, 120960, 120960, 120960, 3628800
Offset: 0

Views

Author

Paul Curtz, Dec 01 2013

Keywords

Comments

A002657(n) and A091137(n) are linked to the Bernoulli numbers B n.
Unreduced differences table of A002657(n)/A091137(n):
1, 1/2, 5/12, 9/24, 251/720, 475/1440,...
-1/2, -1/12, -1/24, -19/720, -27/1440,... =-A141417(n+1)/A091137(n+1),
5/12, 1/24, 11/720, 11/1440,...
-9/24, -19/720, -11/1440,...
251/720, 27/1440,...
-475/1440,... etc.
This is an autosequence of the second kind: its inverse binomial transform is the signed sequence with the main diagonal double of the first upper diagonal.
a(n) is the denominators written by antidiagonals.

Examples

			1,
2,   2,
12, 12, 12,
24, 24, 24, 24, etc.

References

  • P. Curtz, Intégration numérique des systèmes différentiels à conditions initiales. Centre de Calcul Scientifique de l'Armement, Arcueil 1969. Pages 36 and 56.

Crossrefs

Cf. A195287, A002208/A002209 (reduced autosequence), A193546, A174727, A165313.

Formula

Repeat n+1 times A091137(n). Triangle.

A176710 Mix A001021, 2*A001021.

Original entry on oeis.org

1, 2, 12, 24, 144, 288, 1728, 3456, 20736, 41472, 248832, 497664, 2985984, 5971968, 35831808, 71663616, 429981696, 859963392, 5159780352, 10319560704, 61917364224, 123834728448, 743008370688, 1486016741376, 8916100448256, 17832200896512, 106993205379072, 213986410758144
Offset: 0

Views

Author

Paul Curtz, Apr 24 2010

Keywords

Comments

Divides A091137. See A165313.

Links

Programs

  • Mathematica
    LinearRecurrence[{0, 12}, {1, 2}, 30] (* Paolo Xausa, Jan 29 2024 *)

Formula

a(n)= +12*a(n-2). G.f.: ( -1-2*x ) / ( -1+12*x^2 ). [R. J. Mathar, Apr 26 2010]

Extensions

a(21)-a(27) from Paolo Xausa, Jan 29 2024
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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