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A047211
Numbers that are congruent to {2, 4} mod 5.
35
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
1,1
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
LINKS
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].
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
EXAMPLE
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
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 *)
PROG
(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
(PARI) a(n)=(5*n-1)\2 \\ Charles R Greathouse IV, Sep 24 2015
CROSSREFS
Cf. A053685 (subsequence).
Sequence in context: A329846 A067839 A329991 * A225000 A189677 A087733
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Dec 11 1999
EXTENSIONS
Conjecture corrected by John M. Campbell, Aug 25 2011
STATUS
approved