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 1-5 of 5 results.

A047336 Numbers that are congruent to {1, 6} mod 7.

Original entry on oeis.org

1, 6, 8, 13, 15, 20, 22, 27, 29, 34, 36, 41, 43, 48, 50, 55, 57, 62, 64, 69, 71, 76, 78, 83, 85, 90, 92, 97, 99, 104, 106, 111, 113, 118, 120, 125, 127, 132, 134, 139, 141, 146, 148, 153, 155, 160, 162, 167, 169, 174, 176, 181, 183, 188, 190, 195, 197, 202, 204, 209
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Cf. property described by Gary Detlefs in A113801: more generally, these numbers are of the form (2*h*n+(h-4)*(-1)^n-h)/4 (h, n natural numbers), therefore ((2*h*n+(h-4)*(-1)^n-h)/4)^2-1 == 0 (mod h); in this case, a(n)^2-1 == 0 (mod 7). - Bruno Berselli, Nov 17 2010

Links

Crossrefs

Cf. A007310, A019674, A047522, A045472 (primes), A195041 (partial sums), A005408, A047209, A056020, A090771, A091998, A113801, A160389, A175885, A175886, A175887, A178818, A371858.

Programs

  • Haskell
    a047336 n = a047336_list !! (n-1)
a047336_list = 1 : 6 : map (+ 7) a047336_list
-- Reinhard Zumkeller, Jan 07 2012
  • Magma
    [n: n in [1..210]| n mod 7 in {1,6}]; // Bruno Berselli, Feb 22 2011
  • Mathematica
    Rest[Flatten[Table[{7i-1,7i+1},{i,0,40}]]] (* Harvey P. Dale, Nov 20 2010 *)
  • PARI
    a(n)=n\2*7-(-1)^n \\ Charles R Greathouse IV, May 02 2016

Formula

a(1) = 1; a(n) = 7(n-1) - a(n-1). - Rolf Pleisch, Jan 31 2008 (corrected by Jon E. Schoenfield, Dec 22 2008)
a(n) = (7/2)*(n-(1-(-1)^n)/2) - (-1)^n. - Rolf Pleisch, Nov 02 2010
From Bruno Berselli, Nov 17 2010: (Start)
G.f.: x*(1+5*x+x^2)/((1+x)*(1-x)^2).
a(n) = -a(-n+1) = a(n-1) + a(n-2) - a(n-3).
a(n) = a(n-2)+7.
a(n) = 7*A000217(n-1)+1 - 2*Sum_{i=1..n-1} a(i) for n > 1. (End)
a(n) = 7*floor(n/2)+(-1)^(n+1). - Gary Detlefs, Dec 29 2011
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/7)*cot(Pi/7) = A019674 * A178818. - Amiram Eldar, Dec 04 2021
E.g.f.: 1 + ((14*x - 7)*exp(x) + 3*exp(-x))/4. - David Lovler, Sep 01 2022
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 2*cos(Pi/7) (A160389).
Product_{n>=2} (1 + (-1)^n/a(n)) = (Pi/7) * cosec(Pi/7) (A371858). (End)

Extensions

More terms from Jon E. Schoenfield, Jan 18 2009

A019695 Decimal expansion of 2*Pi/7.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Equals twice A019674.

Programs

A195056 Decimal expansion of Pi^2/7.

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Oct 04 2011

Keywords

Examples

			1.409943485869908374119212999982307305045...

References

  • F. Aubonnet, D. Guinin and B.Joppin, Précis de Mathématiques, Analyse 2, Classes Préparatoires, Premier Cycle Universitaire, Bréal, 1990, Exercice 908, pages 82 and 91-92.

Links

Crossrefs

Cf. A000265, A002388, A013661, A019674, A100044, A111003, A164102.

Programs

  • Magma
    Pi(RealField(128))^2/7; // G. C. Greubel, Jun 02 2021
  • Mathematica
    RealDigits[Pi^2/7, 10, 105][[1]] (* T. D. Noe, Oct 05 2011 *)
  • PARI
    Pi^2/7 \\ Michel Marcus, Feb 04 2022
  • Sage
    numerical_approx(pi^2/7, digits=128) # G. C. Greubel, Jun 02 2021

Formula

Equals Sum_{k>=1} A000265(k)/k^3. - Amiram Eldar, Jun 27 2020
Equals Integral_{x=0..1} log(1+x+x^2+x^3+x^4+x^5+x^6)/x dx (Aubonnet). - Bernard Schott, Feb 04 2022

Extensions

Extended by T. D. Noe, Oct 05 2011

A132700 Decimal expansion of Pi/31.

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Aug 27 2007

Keywords

Examples

			Pi/31 = 0.1013416985...

Links

Crossrefs

Cf. A000796, A019674, A132721.

Programs

Extensions

Terms a(30) and beyond from Andrew Howroyd, Jan 03 2020

A371858 Decimal expansion of Integral_{x=0..oo} 1 / (1 + x^7) dx.

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Apr 09 2024

Keywords

Examples

			1.0343760552667964829453064065124887483642567...

Links

Crossrefs

Decimal expansion of Integral_{x=0..oo} 1 / (1 + x^k) dx: A019669 (k=2), A248897 (k=3), A093954 (k=4), A352324 (k=5), A019670 (k=6), this sequence (k=7), A352125 (k=8).
Cf. A019674, A047336, A121598.

Programs

  • Mathematica
    RealDigits[(1/7) Pi Csc[Pi/7], 10, 102][[1]]

Formula

Equals (1/7) * Pi * csc(Pi/7).
Equals A019674 * A121598.
Equals Product_{k>=2} (1 + (-1)^k/A047336(k)). - Amiram Eldar, Nov 22 2024
Showing 1-5 of 5 results.

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

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