A002081 Numbers congruent to {2, 4, 8, 16} (mod 20).

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: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

First differences are periodic, cf. A000689.

References

• 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

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• C. Babbage, On the Determination of the General Term of a New Class of Infinite Series, Trans. Camb. Phil. Soc., 2 (1827), 217-225 (see p. 220).
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Crossrefs

Cf. A002082, A008587.

Programs

Haskell

a002081 n = a002081_list
a002081_list = filter ((`elem` [2,4,8,16]) . (`mod` 20)) [1..]
-- Reinhard Zumkeller, Sep 15 2011

Maple

A002081:=2*(1+2*z**2+2*z**3)/(z**2+1)/(z-1)**2; # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

Flatten[Table[20n + {2, 4, 8, 16}, {n, 0, 14}]] (* Alonso del Arte, Nov 30 2011 *)
LinearRecurrence[{2, -2, 2, -1},{2, 4, 8, 16},57] (* Ray Chandler, Aug 25 2015 *)
Select[Range[300],MemberQ[{2,4,8,16},Mod[#,20]]&] (* Harvey P. Dale, Jul 20 2021 *)

PARI

a(n) = 5*n + [2,-1,-2,1][(n%4)+1] \\ Ralf Stephan, Jun 08 2005

PARI

is(n) = n > 0 && setsearch([2,4,8,16], n%20) > 0 \\ Rick L. Shepherd, Aug 17 2016

Formula

G.f.: 2*(1+2*x^2+2*x^3)/((1-x)^2*(1+x^2)). - Simon Plouffe

a(n+4) = a(n) + 20 for n > 3. - Reinhard Zumkeller, Sep 15 2011

a(n) = 5*n + (1/2)*(3 + (-1)^n)*(-1)^(n(n+1)/2). - Bruno Berselli, Sep 15 2011

E.g.f.: 2*cos(x) - sin(x) + 5*x*exp(x). - Ilya Gutkovskiy, Aug 17 2016

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Jul 31 2000