A047211 Numbers that are congruent to {2, 4} mod 5.

2, 4, 7, 9, 12, 14, 17, 19, 22, 24, 27, 29, 32, 34, 37, 39, 42, 44, 47, 49, 52, 54, 57, 59, 62, 64, 67, 69, 72, 74, 77, 79, 82, 84, 87, 89, 92, 94, 97, 99, 102, 104, 107, 109, 112, 114, 117, 119, 122, 124, 127, 129, 132, 134, 137, 139, 142, 144, 147, 149, 152, 154, 157, 159, 162, 164, 167, 169, 172, 174, 177, 179, 182, 184

Offset: 1

Author

N. J. A. Sloane, Dec 11 1999

Keywords

• nonn
• easy

Comments

Conjecture: n such that the characteristic polynomial of M(n) is irreducible over the rationals where M(n) is an n X n matrix with ones on the skew diagonal and below it and the skew line two positions above it and otherwise zeros; see example for one such matrix. Tested up to n=177. - Joerg Arndt, Aug 10 2011

Examples

			The 7 X 7 matrix (dots for zeros):
[....1.1]
[...1.11]
[..1.111]
[.1.1111]
[1.11111]
[.111111]
[1111111]
has the characteristic polynomial x^7 - 5*x^6 - 4*x^5 + 15*x^4 + 5*x^3 - 11*x^2 - x + 1 which is irreducible over the field of rational numbers, and 7 is a term of the sequence. - _Joerg Arndt_, Aug 10 2011
		

Links

• David Lovler, Table of n, a(n) for n = 1..1000
• Melvyn B. Nathanson, On the fractional parts of roots of positive real numbers, Amer. Math. Monthly, Vol. 120, No. 5 (2013), pp. 409-429 [see p. 417].
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A001622, A047209.

Cf. A053685 (subsequence).

Programs

Haskell

a047211 n = a047211_list !! (n-1)
a047211_list = filter ((`elem` [2,4]) . (`mod` 5)) [1..]
-- Reinhard Zumkeller, Oct 03 2012

Magma

[Floor((5*n-1)/2): n in [1..50]]; // Wesley Ivan Hurt, May 25 2014

Maple

seq(5*floor((n-1)/2) +3 +(-1)^n, n=1..50); # Gary Detlefs, Mar 02 2010

Mathematica

Select[Range[0, 200], MemberQ[{2, 4}, Mod[#, 5]] &] (* Vladimir Joseph Stephan Orlovsky, Feb 12 2012 *)
LinearRecurrence[{1,1,-1},{2,4,7},80] (* Harvey P. Dale, Mar 26 2024 *)

PARI

a(n)=(5*n-1)\2 \\ Charles R Greathouse IV, Sep 24 2015

Formula

a(n) = a(n-1) +a(n-2) -a(n-3).

a(n) = (10*n-3-(-1)^n)/4, (n>=1). [Corrected by Bruno Berselli, Sep 20 2010]

a(n) = 5*floor((n-1)/2) +3 +(-1)^n. - Gary Detlefs, Mar 02 2010

G.f.: x*(2+2*x+x^2)/((1+x)*(1-x)^2). - Paul Barry, Sep 11 2008

a(n) = 5*n-a(n-1)-4 (with a(1)=2). - Vincenzo Librandi, Nov 18 2010

a(n) = floor((5*n-1)/2). - Gary Detlefs, May 14 2011

a(n) = 2*n + floor((n-1)/2). - Arkadiusz Wesolowski, Sep 19 2012

Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(2+2/sqrt(5))*Pi/10 - sqrt(5)*log(phi)/5, where phi is the golden ratio (A001622). - Amiram Eldar, Dec 07 2021

E.g.f.: 1 + ((10*x - 3)*exp(x) - exp(-x))/4. - David Lovler, Aug 23 2022

Extensions

Conjecture corrected by John M. Campbell, Aug 25 2011