A016957 a(n) = 6*n + 4.

4, 10, 16, 22, 28, 34, 40, 46, 52, 58, 64, 70, 76, 82, 88, 94, 100, 106, 112, 118, 124, 130, 136, 142, 148, 154, 160, 166, 172, 178, 184, 190, 196, 202, 208, 214, 220, 226, 232, 238, 244, 250, 256, 262, 268, 274, 280, 286, 292, 298, 304, 310, 316, 322, 328

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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 to sequences related to polygonal numbers.
• Index entries for linear recurrences with constant coefficients, signature (2,-1).

Crossrefs

Cf. A000217, A005408, A006370, A008588, A008615, A016921, A016789, A016933, A016945, A016958, A016969, A017617, A017329, A057145, A067412, A088827, A139600, A139606, A157176, A186424.

Programs

Haskell

a016957 = (+ 4) . (* 6)  -- Reinhard Zumkeller, Jul 05 2013

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 *)

Maxima

makelist(6*n+4, n, 0, 30); /* Martin Ettl, Nov 12 2012 */

PARI

a(n)=6*n+4 \\ Charles R Greathouse IV, Jul 10 2016

Formula

A008615(a(n)) = n+1. - Reinhard Zumkeller, Feb 27 2008

a(n) = A016789(n)*2. - Omar E. Pol, May 02 2008

A157176(a(n)) = A067412(n+1). - Reinhard Zumkeller, Feb 24 2009

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