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%I #88 Mar 21 2024 08:37:13
%S 2,8,14,20,26,32,38,44,50,56,62,68,74,80,86,92,98,104,110,116,122,128,
%T 134,140,146,152,158,164,170,176,182,188,194,200,206,212,218,224,230,
%U 236,242,248,254,260,266,272,278,284,290,296,302,308,314,320,326
%N a(n) = 6*n + 2.
%C Number of 3 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (10;0) 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 same relative order as those in the triple (x,y,z). - _Sergey Kitaev_, Nov 11 2004
%C Exponents n>1 for which 1 - x + x^n is reducible. - _Ron Knott_, Oct 13 2016
%C For the Collatz problem, these are the descenders' values that require division by 2. - _Fred Daniel Kline_, Jan 19 2017
%C For n > 3, also the number of (not necessarily maximal) cliques in the n-helm graph. - _Eric W. Weisstein_, Nov 29 2017
%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 Sergey Kitaev, <a href="https://web.archive.org/web/20130625171839/http://www.ms.uky.edu/~math/MAreport/4-ser.ps">On multi-avoidance of right angled numbered polyomino patterns</a>, University of Kentucky Research Reports (2004).
%H Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Clique.html">Clique</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HelmGraph.html">Helm Graph</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 A157176(a(n)) = A013730(n). - _Reinhard Zumkeller_, Feb 24 2009
%F A089911(2*a(n)) = 3. - _Reinhard Zumkeller_, Jul 05 2013
%F a(n) = 2*(6*n-1) - a(n-1) (with a(0)=2). - _Vincenzo Librandi_, Nov 20 2010
%F G.f.: 2*(1+2*x)/(1-x)^2. - _Colin Barker_, Jan 08 2012
%F a(n) = (3 * A016813(n) + 1) / 2.- _Fred Daniel Kline_, Jan 20 2017
%F a(n) = A016789(A005843(n)). - _Felix Fröhlich_, Jan 20 2017
%F Sum_{n>=0} (-1)^n/a(n) = sqrt(3)*Pi/18 + log(2)/6. - _Amiram Eldar_, Dec 10 2021
%F a(n) = 2 * A016777(n). - _Alois P. Heinz_, Dec 27 2023
%F From _Elmo R. Oliveira_, Mar 08 2024: (Start)
%F a(n) = 2*a(n-1) - a(n-2) for n >= 2.
%F E.g.f.: 2*exp(x)*(1 + 3*x). (End)
%p a[1]:=2:for n from 2 to 100 do a[n]:=a[n-1]+6 od: seq(a[n], n=1..47); # _Zerinvary Lajos_, Mar 16 2008
%t Range[2, 500, 6] (* _Vladimir Joseph Stephan Orlovsky_, May 26 2011 *)
%t Table[6 n + 2, {n, 0, 20}] (* _Eric W. Weisstein_, Nov 29 2017 *)
%t 6 Range[0, 20] + 2 (* _Eric W. Weisstein_, Nov 29 2017 *)
%t LinearRecurrence[{2, -1}, {8, 14}, {0, 20}] (* _Eric W. Weisstein_, Nov 29 2017 *)
%t CoefficientList[Series[2 (1 + 2 x)/(-1 + x)^2, {x, 0, 20}], x] (* _Eric W. Weisstein_, Nov 29 2017 *)
%o (Sage) [i+2 for i in range(280) if gcd(i,6) == 6] # _Zerinvary Lajos_, May 20 2009
%o (Haskell)
%o a016933 = (+ 2) . (* 6) -- _Reinhard Zumkeller_, Jul 05 2013
%o (PARI) a(n)=6*n+2 \\ _Charles R Greathouse IV_, Jul 10 2016
%Y Cf. A008588, A016921, A016945, A016957, A016969, A017569, A016777, A016813.
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_
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