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

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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

A178817 Decimal expansion of the area of the regular 7-gon (heptagon) of edge length 1.

Original entry on oeis.org

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

Views

Author

Vladimir Joseph Stephan Orlovsky, Jun 16 2010

Keywords

Examples

			3.63391244400158899253619300760022057873501036159544491714598040951029...

Links

Crossrefs

Cf. A178818, A343058.
Cf. Areas of other regular polygons: A120011, A102771, A104956, A090488, A256853, A178816, A256854, A178809.

Programs

  • Magma
    SetDefaultRealField(RealField(100)); R:=RealField(); 7*Cot(Pi(R)/7)/4; // G. C. Greubel, Jan 22 2019
  • Maple
    evalf[120]((7/4)*cot(Pi/7)); # Muniru A Asiru, Jan 22 2019
  • Mathematica
    RealDigits[7*Cot[Pi/7]/4, 10, 100][[1]]
  • PARI
    p=7; a=(p/4)*cotan(Pi/p)  \\ Set realprecision in excess. - Stanislav Sykora, Apr 12 2015
  • Sage
    numerical_approx(7*cot(pi/7)/4, digits=100) # G. C. Greubel, Jan 22 2019

Formula

Equals (7/4) * cot(Pi/7).
From Michal Paulovic, Dec 27 2022: (Start)
Equals 7 / (4 * tan(Pi/7)) = 7 / (4 * A343058).
Equals sqrt(7/3 * (35 + 2 * 196^(1/3) * ((13 - 3 * sqrt(3) * i)^(1/3) + (13 + 3 * sqrt(3) * i)^(1/3)))) / 4.
Equals sqrt(7/4) * sqrt(35/12 + (637/54 - sqrt(-2401/108))^(1/3) + (637/54 + sqrt(-2401/108))^(1/3)).
(End)
A root of the polynomial 4096*x^6 - 62720*x^4 + 115248*x^2 - 16807. - Joerg Arndt, Jan 02 2023

A374971 Decimal expansion of the apothem (inradius) of a regular heptagon with unit side length.

Original entry on oeis.org

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

Views

Author

Paolo Xausa, Jul 26 2024

Keywords

Examples

			1.0382606982861682835817694307429201653528601033...

Links

Crossrefs

Cf. A374957 (circumradius), A374972 (sagitta), A178817 (area).
Cf. apothem of other polygons with unit side length: A020769 (triangle), A020761 (square), A375067 (pentagon), A010527 (hexagon), A174968 (octagon), A375152 (9-gon), A179452 (10-gon), A375191 (11-gon), A375193 (12-gon).
Cf. A178818, A073052, A343058.

Programs

Formula

Equals cot(Pi/7)/2 = A178818/2.
Equals 1/(2*tan(Pi/7)) = 1/(2*A343058).
Equals A374957*cos(Pi/7) = A374957*A073052.
Equals A374957 - A374972.
Largest of the 6 real-valued roots of 448*x^6 -560*x^4 +84*x^2 -1 =0. - R. J. Mathar, Aug 29 2025

A343059 Decimal expansion of tan(Pi/14).

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, Apr 04 2021

Keywords

Comments

Root of the equation -1 + 21*x^2 - 35*x^4 + 7*x^6 = 0. (Other: A178818) - Vaclav Kotesovec, Apr 04 2021

Examples

			0.228243474390149938077611362061014782...

Links

Crossrefs

Cf. A232736 (sin(Pi/14)), A232735 (cos(Pi/14)).

Programs

  • Mathematica
    RealDigits[Tan[Pi/14], 10, 125][[1]] (* Amiram Eldar, Apr 27 2021 *)
  • PARI
    tan(Pi/14)
  • SageMath
    numerical_approx(tan(pi/14), digits=124) # G. C. Greubel, Sep 30 2022

Formula

Equals A323601/(1+A073052). - R. J. Mathar, Sep 06 2025

A216606 Decimal expansion of 360/7.

Original entry on oeis.org

5, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7
Offset: 2

Views

Author

Paul Curtz, Sep 10 2012

Keywords

Comments

A020806 preceded by a 5.
Number of degrees in the exterior angle of an equilateral heptagon. Since 1969, used in many (orbiform or Reuleaux) heptagonal coins. Zambia has a natural heptagonal coin. Brazil and Costa Rica have a coin with the natural heptagon inscribed in the coin's disk.

Examples

			51.42857...

Links

Crossrefs

Cf. A019695, A020806, A021018, A060708, A068028, A178817, A178818.

Programs

Formula

a(n) = 50 + 10*A020806(n).
After 5, of period 6: repeat [1, 4, 2, 8, 5, 7].
From Wesley Ivan Hurt, Jun 28 2016: (Start)
G.f.: x^3*(5-4*x+3*x^2+3*x^3+2*x^4) / (1-x+x^3-x^4).
a(n) = 9/2 + 11*cos(n*Pi)/6 + 5*cos(n*Pi/3)/3 + sqrt(3)*sin(n*Pi/3), n>2.
a(n) = a(n-1) - a(n-3) + a(n-4) for n>6, a(n) = a(n-6) for n>8. (End)
Showing 1-5 of 5 results.

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

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