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%I #126 Jul 11 2022 08:36:32
%S 4,10,16,22,28,34,40,46,52,58,64,70,76,82,88,94,100,106,112,118,124,
%T 130,136,142,148,154,160,166,172,178,184,190,196,202,208,214,220,226,
%U 232,238,244,250,256,262,268,274,280,286,292,298,304,310,316,322,328
%N a(n) = 6*n + 4.
%C 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
%C 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
%C 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
%C 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
%C 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
%C 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
%C 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
%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189. - From _N. J. A. Sloane_, Dec 01 2012
%H Heinz Ebert, <a href="https://arxiv.org/abs/1905.07575">A Graph Theoretical Approach to the Collatz Problem</a>, arXiv:1905.07575 [math.GM], 2019-2020.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.
%H Sergey Kitaev, <a href="http://www.emis.de/journals/INTEGERS/papers/e21/e21.Abstract.html">On multi-avoidance of right angled numbered polyomino patterns</a>, Integers: Electronic Journal of Combinatorial Number Theory, Vol. 4 (2004), Article A21, 20pp.
%H <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F A008615(a(n)) = n+1. - _Reinhard Zumkeller_, Feb 27 2008
%F a(n) = A016789(n)*2. - _Omar E. Pol_, May 02 2008
%F A157176(a(n)) = A067412(n+1). - _Reinhard Zumkeller_, Feb 24 2009
%F a(n) = sqrt(A016958(n)). - _Zerinvary Lajos_, Jun 30 2009
%F a(n) = 2*(6*n+1) - a(n-1) (with a(0)=4). - _Vincenzo Librandi_, Nov 20 2010
%F a(n) = floor((sqrt(36*n^2 - 36*n + 1) + 6*n + 1)/2). - _Srikanth K S_, Dec 18 2010
%F From _Colin Barker_, Jan 30 2012: (Start)
%F G.f.: 2*(2+x)/(1-2*x+x^2).
%F a(n) = 2*a(n-1) - a(n-2). (End)
%F A089911(2*a(n)) = 9. - _Reinhard Zumkeller_, Jul 05 2013
%F a(n) = 3 * A005408(n) + 1. - _Fred Daniel Kline_, Oct 24 2015
%F a(n) = A057145(n+2,4). - _R. J. Mathar_, Jul 28 2016
%F a(4*n+2) = 4 * a(n). - _Zhandos Mambetaliyev_, Sep 22 2018
%F Sum_{n>=0} (-1)^n/a(n) = sqrt(3)*Pi/18 - log(2)/6. - _Amiram Eldar_, Dec 10 2021
%p seq(6*n+4, n = 0 .. 50) # _Matt C. Anderson_, Jun 09 2017
%t Range[4, 1000, 6] (* _Vladimir Joseph Stephan Orlovsky_, May 27 2011 *)
%o (Maxima) makelist(6*n+4, n, 0, 30); /* _Martin Ettl_, Nov 12 2012 */
%o (Haskell)
%o a016957 = (+ 4) . (* 6) -- _Reinhard Zumkeller_, Jul 05 2013
%o (PARI) a(n)=6*n+4 \\ _Charles R Greathouse IV_, Jul 10 2016
%Y Cf. A006370, A008588, A016921, A016933, A016945, A016969, A000217, A017329, A057145, A139600, A139606, A016958, A186424.
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_
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