|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
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
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
|
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..]
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|