OFFSET
0,1
COMMENTS
Number of 2 X n binary matrices avoiding simultaneously the right-angled numbered polyomino patterns (ranpp) (00;1), (01,1) and (11;0). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1 < i2, j1 < j2 and these elements are in the same relative order as those in the triple (x,y,z). In general, the number of m X n 0-1 matrices in question is given by (n+2)*2^(m-1) + 2*m*(n-1) - 2 for m > 1 and n > 1. - Sergey Kitaev, Nov 12 2004
If Y is a 4-subset of an n-set X then, for n >= 4, a(n-4) is the number of 3-subsets of X having at least two elements in common with Y. - Milan Janjic, Dec 08 2007
4th transversal numbers (or 4-transversal numbers): Numbers of the 4th column of positive numbers in the square array of nonnegative and polygonal numbers A139600. Also, numbers of the 4th column in the square array A057145. - Omar E. Pol, May 02 2008
a(n) is the maximum number such that there exists an edge coloring of the complete graph with a(n) vertices using n colors and every subgraph whose edges are of the same color (subgraph induced by edge color) is planar. - Srikanth K S, Dec 18 2010
Also numbers having two antecedents in the Collatz problem: 12*n+8 and 2*n+1 (respectively A017617(n) and A005408(n)). - Michel Lagneau, Dec 28 2012
a(n) = 6n+4 has three undirected edges e1 = (3n+2, 6n+4), e2 = (6n+4, 12n+8) and e3 = (2n+1, 6n+4) in the Collatz graph of A006370. - Heinz Ebert, Mar 16 2021
Conjecture: this sequence contains some but not all, even numbers with odd abundance A088827. They appear in this sequence at indices A186424(n) - 1. - John Tyler Rascoe, Jul 09 2022
REFERENCES
A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189. - From N. J. A. Sloane, Dec 01 2012
LINKS
Heinz Ebert, A Graph Theoretical Approach to the Collatz Problem, arXiv:1905.07575 [math.GM], 2019-2020.
Tanya Khovanova, Recursive Sequences.
Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory, Vol. 4 (2004), Article A21, 20pp.
Index entries for linear recurrences with constant coefficients, signature (2,-1).
FORMULA
A008615(a(n)) = n+1. - Reinhard Zumkeller, Feb 27 2008
a(n) = A016789(n)*2. - Omar E. Pol, May 02 2008
a(n) = sqrt(A016958(n)). - Zerinvary Lajos, Jun 30 2009
a(n) = 2*(6*n+1) - a(n-1) (with a(0)=4). - Vincenzo Librandi, Nov 20 2010
a(n) = floor((sqrt(36*n^2 - 36*n + 1) + 6*n + 1)/2). - Srikanth K S, Dec 18 2010
From Colin Barker, Jan 30 2012: (Start)
G.f.: 2*(2+x)/(1-2*x+x^2).
a(n) = 2*a(n-1) - a(n-2). (End)
A089911(2*a(n)) = 9. - Reinhard Zumkeller, Jul 05 2013
a(n) = 3 * A005408(n) + 1. - Fred Daniel Kline, Oct 24 2015
a(n) = A057145(n+2,4). - R. J. Mathar, Jul 28 2016
a(4*n+2) = 4 * a(n). - Zhandos Mambetaliyev, Sep 22 2018
Sum_{n>=0} (-1)^n/a(n) = sqrt(3)*Pi/18 - log(2)/6. - Amiram Eldar, Dec 10 2021
E.g.f.: 2*exp(x)*(2 + 3*x). - Stefano Spezia, May 29 2024
MAPLE
seq(6*n+4, n = 0 .. 50) # Matt C. Anderson, Jun 09 2017
MATHEMATICA
Range[4, 1000, 6] (* Vladimir Joseph Stephan Orlovsky, May 27 2011 *)
PROG
(Maxima) makelist(6*n+4, n, 0, 30); /* Martin Ettl, Nov 12 2012 */
(Haskell)
a016957 = (+ 4) . (* 6) -- Reinhard Zumkeller, Jul 05 2013
(PARI) a(n)=6*n+4 \\ Charles R Greathouse IV, Jul 10 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved