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.

A000689 Final decimal digit of 2^n.

Original entry on oeis.org

1, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

These are the analogs of the powers of 2 in carryless arithmetic mod 10.
Let G = {2,4,8,6}. Let o be defined as XoY = least significant digit in XY. Then (G,o) is an Abelian group wherein 2 is a generator (also see the first comment under A001148). - K.V.Iyer, Mar 12 2010
This is also the decimal expansion of 227/1818. - Kritsada Moomuang, Dec 21 2021

Examples

			G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 6*x^4 + 2*x^5 + 4*x^6 + 8*x^7 + 6*x^8 + ...

Links

Crossrefs

Cf. A000079, A000855, A008952, A034887, A173635.

Programs

  • Haskell
    a000689 n = a000689_list !! n
a000689_list = 1 : cycle [2,4,8,6]  -- Reinhard Zumkeller, Sep 15 2011
  • Magma
    [2^n mod 10: n in [0..150]]; // Vincenzo Librandi, Apr 12 2011
  • Mathematica
    Table[PowerMod[2, n, 10], {n, 0, 200}] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)
  • PARI
    for(n=0,80, if(n,{x=(n+3)%4+1; print1(10-(4*x^3+47*x-27*x^2)/3,", ")},{print1("1, ")}))
  • SageMath
    [power_mod(2,n,10)for n in range(0, 81)] # Zerinvary Lajos, Nov 03 2009

Formula

Periodic with period 4.
a(n) = 2^n mod 10.
a(n) = A002081(n) - A002081(n-1), for n > 0.
From R. J. Mathar, Apr 13 2010: (Start)
a(n) = a(n-1) - a(n-2) + a(n-3), n > 3.
G.f.: (x+3*x^2+5*x^3+1)/((1-x) * (1+x^2)). (End)
For n >= 1, a(n) = 10 - (4x^3 + 47x - 27x^2)/3, where x = (n+3) mod 4 + 1.
For n >= 1, a(n) = A070402(n) + 5*floor( ((n-1) mod 4)/2 ).
G.f.: 1 / (1 - 2*x / (1 + 5*x^3 / (1 + x / (1 - 3*x / (1 + 3*x))))). - Michael Somos, May 12 2012
a(n) = 5 + cos((n*Pi)/2) - 3*sin((n*Pi)/2) for n >= 1. - Kritsada Moomuang, Dec 21 2021

A102039 a(n) = a(n-1) + last digit of a(n-1), starting at 1.

Original entry on oeis.org

1, 2, 4, 8, 16, 22, 24, 28, 36, 42, 44, 48, 56, 62, 64, 68, 76, 82, 84, 88, 96, 102, 104, 108, 116, 122, 124, 128, 136, 142, 144, 148, 156, 162, 164, 168, 176, 182, 184, 188, 196, 202, 204, 208, 216, 222, 224, 228, 236, 242, 244, 248, 256, 262, 264, 268, 276, 282
Offset: 1

Views

Author

Samantha Stones (devilsdaughter2000(AT)hotmail.com), Dec 25 2004

Keywords

Comments

Sequence A001651 is the "base 3" version. In base 4 this rule leads to (1,2,4,4,4...), in base 5 to (1,2,4,8,11,12,14,18,21,22,24,28...) = A235700. - M. F. Hasler, Jan 14 2014
This and the following sequences (none of which is "base"!) could all be defined by a(1) = 1 and a(n+1) = a(n) + (a(n) mod b) with different values of b: A001651 (b=3), A235700 (b=5), A047235 (b=6), A047350 (b=7), A007612 (b=9). Using b=4 or b=8 yields a constant sequence from that term on. - M. F. Hasler, Jan 15 2014

Examples

			28 + 8 = 36, 36 + 6 = 42.

Links

Crossrefs

Apart from initial term, same as A002081.

Programs

  • Mathematica
    LinearRecurrence[{2,-2,2,-1},{1,2,4,8,16},60] (* Harvey P. Dale, Jul 02 2022 *)
  • PARI
    print1(a=1);for(i=1,99,print1(","a+=a%10)) \\ M. F. Hasler, Jan 14 2014
  • PARI
    Vec(x*(5*x^4+2*x^3+2*x^2+1)/((x-1)^2*(x^2+1)) + O(x^100)) \\ Colin Barker, Sep 20 2014
  • PARI
    a(n) = if(n==1, 1, (-10-(1-I/2)*(-I)^n-(1+I/2)*I^n+5*n)) \\ Colin Barker, Oct 18 2015

Formula

a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-a(n-4) for n>5. G.f.: x*(5*x^4+2*x^3+2*x^2+1) / ((x-1)^2*(x^2+1)). - Colin Barker, Sep 20 2014
a(n) = (-10 - (1-i/2)*(-i)^n - (1+i/2)*i^n + 5*n) for n>1, where i = sqrt(-1). - Colin Barker, Oct 18 2015

A002082 2nd differences are periodic.

Original entry on oeis.org

2, 2, 4, 10, 16, 28, 48, 76, 110, 144, 182, 222, 264, 310, 356, 408, 468, 536, 610, 684, 762, 842, 924, 1010, 1096, 1188, 1288, 1396, 1510, 1624, 1742, 1862, 1984, 2110, 2236, 2368
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • "On the Determination of the General Term of a New Class of Infinite Series", by Charles Babbage, Trans. Camb. Phil. Soc., 2 (1827), 217-225 (see p. 220).
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002081.

Programs

  • Mathematica
    LinearRecurrence[{3, -4, 4, -4, 4, -4, 4, -4, 4, -3, 1},{2, 2, 4, 10, 16, 28, 48, 76, 110, 144, 182},36] (* Ray Chandler, Aug 25 2015 *)

Formula

G.f.: (-4x^10+2x^9-2x^8-4x^7-4x^6-4x^5-2x^4+2x^3-6x^2+4x-2)/((x^4-x^3+x^2-x+1)*(x^4+x^3+x^2+x+1)*(x-1)^3). - Ralf Stephan, Mar 11 2003
Showing 1-3 of 3 results.

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

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