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 1-3 of 3 results.

A001148 Final digit of 3^n.

Original entry on oeis.org

1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Let G = {1,3,7,9}, and let the binary operator o be defined as: X o Y = least significant digit of the product XY, where X,Y belong to G. Then (G,o) is an Abelian group and 3 is a generator of this group. - K.V.Iyer, Apr 19 2009
3^n mod 10 and 3^n mod 20. - Zerinvary Lajos, Nov 25 2009
Continued fraction expansion of (243+17*sqrt(285))/4020 = 0.13183906... (see A178148). - Klaus Brockhaus, Apr 17 2011

Links

Programs

Formula

Periodic with period 4.
From R. J. Mathar, Apr 13 2010: (Start)
a(n) = a(n-1) - a(n-2) + a(n-3).
G.f.: (1+2*x+7*x^2)/ ((1-x) * (1+x^2)). (End)
a(n) = 5 - (2+i)*(-i)^n - (2-i)*i^n, where i is the imaginary unit. Also a(n) = A001903(A159966(n)). - Bruno Berselli, Feb 08 2011
a(0)=1, a(1)=3, a(n) = 10 - a(n-2). - Vincenzo Librandi, Feb 08 2011

A092486 Take natural numbers, exchange first and third quadrisection.

Original entry on oeis.org

3, 2, 1, 4, 7, 6, 5, 8, 11, 10, 9, 12, 15, 14, 13, 16, 19, 18, 17, 20, 23, 22, 21, 24, 27, 26, 25, 28, 31, 30, 29, 32, 35, 34, 33, 36, 39, 38, 37, 40, 43, 42, 41, 44, 47, 46, 45, 48, 51, 50, 49, 52, 55, 54, 53, 56, 59, 58, 57, 60, 63, 62, 61, 64, 67, 66, 65, 68, 71, 70, 69, 72
Offset: 0

Views

Author

Ralf Stephan, Apr 04 2004

Keywords

Links

Crossrefs

Cf. A002162, A080412, A064429, A074066, A080782, A159966.

Programs

  • Mathematica
    Flatten[Partition[Range[80],4]/.{a_,b_,c_,d_}->{c,b,a,d}] (* Harvey P. Dale, Aug 12 2012 *)
  • PARI
    { f="b092486.txt"; for (n=0, 5000, a0=4*n + 3; a1=a0 - 1; a2=a1 - 1; a3=a0 + 1; write(f, 4*n, " ", a0); write(f, 4*n+1, " ", a1); write(f, 4*n+2, " ", a2); write(f, 4*n+3, " ", a3); ); } \\ Harry J. Smith, Jun 21 2009

Formula

G.f.: (3-4*x+3*x^2)/((1+x^2)*(1-x)^2).
a(4n) = 4n+3, a(4n+1) = 4n+2, a(4n+2) = 4n+1, a(4n+3) = 4n+4.
a(n) = n+1+i^n+(-i)^n, where i is the imaginary unit. - Bruno Berselli, Feb 08 2011
From Wesley Ivan Hurt, May 09 2021: (Start)
a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-a(n-4).
a(n) = 1 + n + 2*cos(n*Pi/2). (End)
Sum_{n>=0} (-1)^n/a(n) = log(2) (A002162). - Amiram Eldar, Nov 28 2023

A001903 Final digit of 7^n.

Original entry on oeis.org

1, 7, 9, 3, 1, 7, 9, 3, 1, 7, 9, 3, 1, 7, 9, 3, 1, 7, 9, 3, 1, 7, 9, 3, 1, 7, 9, 3, 1, 7, 9, 3, 1, 7, 9, 3, 1, 7, 9, 3, 1, 7, 9, 3, 1, 7, 9, 3, 1, 7, 9, 3, 1, 7, 9, 3, 1, 7, 9, 3, 1, 7, 9, 3, 1, 7, 9, 3, 1, 7, 9, 3, 1, 7, 9, 3, 1, 7, 9, 3, 1
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Period 4: repeat [1, 7, 9, 3]. - Joerg Arndt, Aug 12 2014

Links

Crossrefs

Cf. A000420, A001148, A010879, A131707, A159966.

Programs

Formula

a(n) = 7^n mod 10. - Zerinvary Lajos, Nov 03 2009
From R. J. Mathar, Apr 20 2010: (Start)
a(n) = a(n-1) - a(n-2) + a(n-3) for n > 2.
G.f.: ( 1+6*x+3*x^2 ) / ( (1-x)*(1+x^2) ). (End)
a(n) = 10 - a(n-2) for n > 1. - Vincenzo Librandi, Feb 08 2011
From Bruno Berselli, Feb 08 2011: (Start)
a(n) = 5 - (2-i)*(-i)^n - (2+i)*i^n, where i=sqrt(-1).
a(n) = A001148(A159966(n)). (End)
a(n) = A010879(A000420(n)). - Michel Marcus, Jul 06 2016
E.g.f.: 2*sin(x) - 4*cos(x) + 5*exp(x). - Ilya Gutkovskiy, Jul 06 2016
Showing 1-3 of 3 results.

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

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