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.

A056020 Numbers that are congruent to +-1 mod 9.

Original entry on oeis.org

1, 8, 10, 17, 19, 26, 28, 35, 37, 44, 46, 53, 55, 62, 64, 71, 73, 80, 82, 89, 91, 98, 100, 107, 109, 116, 118, 125, 127, 134, 136, 143, 145, 152, 154, 161, 163, 170, 172, 179, 181, 188, 190, 197, 199, 206, 208, 215, 217, 224, 226, 233, 235, 242, 244, 251, 253
Offset: 1

Views

Author

Robert G. Wilson v, Jun 08 2000

Keywords

Comments

Or, numbers k such that k^2 == 1 (mod 9).
Or, numbers k such that the iterative cycle j -> sum of digits of j^2 when started at k contains a 1. E.g., 8 -> 6+4 = 10 -> 1+0+0 = 1 and 17 -> 2+8+9 = 19 -> 3+6+1 = 10 -> 1+0+0 = 1. - Asher Auel, May 17 2001

Links

Crossrefs

Cf. A007953, A047522 (n=1 or 7 mod 8), A090771 (n=1 or 9 mod 10).
Cf. A129805 (primes), A195042 (partial sums).
Cf. A005408, A019676, A019968, A047209, A007310, A047336, A175885, A091998, A175886, A113801, A175887, A332437.
Cf. A381319 (general case mod n^2).

Programs

  • Haskell
    a056020 n = a056020_list !! (n-1)
a05602_list = 1 : 8 : map (+ 9) a056020_list
-- Reinhard Zumkeller, Jan 07 2012
  • Mathematica
    Select[ Range[ 300 ], PowerMod[ #, 2, 3^2 ]==1& ]
(* or *)
LinearRecurrence[{1, 1, -1}, {1, 8, 10}, 67] (* Mike Sheppard, Feb 18 2025 *)
  • PARI
    a(n)=9*(n>>1)+if(n%2,1,-1) \\ Charles R Greathouse IV, Jun 29 2011
  • PARI
    for(n=1,40,print1(9*n-8,", ",9*n-1,", ")) \\ Charles R Greathouse IV, Jun 29 2011

Formula

a(1) = 1; a(n) = 9(n-1) - a(n-1). - Rolf Pleisch, Jan 31 2008 [Offset corrected by Jon E. Schoenfield, Dec 22 2008]
From R. J. Mathar, Feb 10 2008: (Start)
O.g.f.: 1 + 5/(4(x+1)) + 27/(4(-1+x)) + 9/(2(-1+x)^2).
a(n+1) - a(n) = A010697(n). (End)
a(n) = (9*A132355(n) + 1)^(1/2). - Gary Detlefs, Feb 22 2010
From Bruno Berselli, Nov 17 2010: (Start)
a(n) = a(n-2) + 9, for n > 2.
a(n) = 9*A000217(n-1) + 1 - 2*Sum_{i=1..n-1} a(i), n > 1. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/9)*cot(Pi/9) = A019676 * A019968. - Amiram Eldar, Dec 04 2021
E.g.f.: 1 + ((18*x - 9)*exp(x) + 5*exp(-x))/4. - David Lovler, Sep 04 2022
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 2*cos(Pi/9) (A332437).
Product_{n>=2} (1 + (-1)^n/a(n)) = (Pi/9)*cosec(Pi/9). (End)
From Mike Sheppard, Feb 18 2025: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3).
a(n) ~ (3^2/2)*n. (End)

A256853 Decimal expansion of the area of a unit 9-gon.

Original entry on oeis.org

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

Views

Author

Stanislav Sykora, Apr 12 2015

Keywords

Comments

From Michal Paulovic, May 09 2024: (Start)
This constant multiplied by the square of the side length of a regular enneagon equals the area of that enneagon.
9^2 divided by this constant equals 36 * tan(Pi/9) = 13.10292843... which is the perimeter and the area of an equable enneagon with its side length 4 * tan(Pi/9) = 1.45588093... . (End)

Examples

			6.181824193772900127213744059619763614941713348134358098386864...

Links

Crossrefs

Cf. A000796, A019669, A019670, A019673, A019676, A019685, A019968, A120011 (p=3), A102771 (p=5), A104956 (p=6), A178817 (p=7), A090488 (p=8), A178816 (p=10), A256854 (p=11), A178809 (p=12).

Programs

  • Maple
    evalf(9 / (4 * tan(Pi/9)), 100); # Michal Paulovic, May 09 2024
  • Mathematica
    RealDigits[(9/4)*Cot[Pi/9], 10, 50][[1]] (* G. C. Greubel, Jul 03 2017 *)
  • PARI
    p=9; a=(p/4)*cotan(Pi/p)        \\ Use realprecision in excess

Formula

Equals (p/4)*cot(Pi/p), with p = 9.
From Michal Paulovic, May 09 2024: (Start)
Equals 9 * sqrt(2 / (1 - sin(5 * A000796 / 18)) - 1) / 4.
Equals 9 * sqrt(2 / (1 - sin(5 * A019669 / 9)) - 1) / 4.
Equals 9 * sqrt(2 / (1 - sin(5 * A019670 / 6)) - 1) / 4.
Equals 9 * sqrt(2 / (1 - sin(5 * A019673 / 3)) - 1) / 4.
Equals 9 * sqrt(2 / (1 - sin(5 * A019676 / 2)) - 1) / 4.
Equals 9 * sqrt(2 / (1 - sin(50 * A019685)) - 1) / 4.
Equals 9 * sqrt(2 / (1 - sin(5 * Pi / 18)) - 1) / 4.
Equals 9 * sqrt(4 / (2 - i^(4/9) - i^(-4/9)) - 1) / 4.
Equals 9 * sqrt(1 / (8 - (-32 + sqrt(-3072))^(1/3) - (-32 - sqrt(-3072))^(1/3)) - 1/16). (End)
Largest of the 6 real-valued roots of 4096*x^6 -186624*x^4 +1154736*x^2 -177147 =0. - R. J. Mathar, Aug 29 2025

A375152 Decimal expansion of the apothem (inradius) of a regular 9-gon with unit side length.

Original entry on oeis.org

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

Views

Author

Paolo Xausa, Aug 01 2024

Keywords

Examples

			1.3737387097273111393808320132488363588759362995854...

Links

Crossrefs

Cf. A375151 (circumradius), A375153 (sagitta), A256853 (area).
Cf. apothem of other polygons with unit side length: A020769 (triangle), A020761 (square), A375067 (pentagon), A010527 (hexagon), A374971 (heptagon), A174968 (octagon), A179452 (10-gon), A375191 (11-gon), A375193 (12-gon).
Cf. A019879, A019918, A019968.

Programs

Formula

Equals cot(Pi/9)/2 = A019968/2.
Equals 1/(2*tan(Pi/9)) = 1/(2*A019918).
Equals A375151*cos(Pi/9) = A375151*A019879.
Equals A375151 - A375153.
Largest of the 6 real-valued roots of 192*x^6 -432*x^4 +132*x^2 -1=0. - R. J. Mathar, Aug 29 2025

A019908 Decimal expansion of tangent of 10 degrees.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Comments

Also the decimal expansion of cotangent of 80 degrees. - Mohammad K. Azarian, Jun 30 2013

Examples

			0.176326980708464973471090386868618986121633...

Links

Crossrefs

Cf. A019918.

Programs

Formula

A root of 3*x^6 -27*x^4 +33*x^2 -1 =0 (others A019968, A019948). - R. J. Mathar, Aug 29 2025
tan(Pi/18) = A019819/A019889. - R. J. Mathar, Aug 31 2025

A019948 Decimal expansion of tangent of 50 degrees.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Comments

Also the decimal expansion of cotangent of 40 degrees. - Ivan Panchenko, Sep 01 2014

Examples

			1.19175359259420995870530807186041933693070040770854385365483...

Links

Crossrefs

Cf. A019859 (sine of 50 degrees).

Programs

  • Magma
    SetDefaultRealField(RealField(100)); R:= RealField(); Tan(5*Pi(R)/18); // G. C. Greubel, Nov 23 2018
  • Mathematica
    RealDigits[Tan[5*Pi/18], 10, 100][[1]] (* G. C. Greubel, Nov 23 2018 *)
  • PARI
    default(realprecision, 100); tan(5*Pi/18) \\ G. C. Greubel, Nov 23 2018
  • Sage
    numerical_approx(tan(5*pi/18), digits=100) # G. C. Greubel, Nov 23 2018

Formula

A root of 3*x^6 -27*x^4 +33*x^2 -1 =0 (others A019968, A019908). - R. J. Mathar, Aug 29 2025
Showing 1-5 of 5 results.

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

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