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%I #121 Jun 25 2023 14:03:18
%S 0,6,12,18,24,30,36,42,48,54,60,66,72,78,84,90,96,102,108,114,120,126,
%T 132,138,144,150,156,162,168,174,180,186,192,198,204,210,216,222,228,
%U 234,240,246,252,258,264,270,276,282,288,294,300,306,312,318,324,330,336,342,348
%N Nonnegative multiples of 6.
%C For n > 3, the number of squares on the infinite 3-column half-strip chessboard at <= n knight moves from any fixed point on the short edge.
%C Second differences of A000578. - Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004
%C A008615(a(n)) = n. - _Reinhard Zumkeller_, Feb 27 2008
%C A157176(a(n)) = A001018(n). - _Reinhard Zumkeller_, Feb 24 2009
%C These numbers can be written as the sum of four cubes (i.e., 6*n = (n+1)^3 + (n-1)^3 + (-n)^3 + (-n)^3). - _Arkadiusz Wesolowski_, Aug 09 2013
%C A122841(a(n)) > 0 for n > 0. - _Reinhard Zumkeller_, Nov 10 2013
%C Surface area of a cube with side sqrt(n). - _Wesley Ivan Hurt_, Aug 24 2014
%C a(n) is representable as a sum of three but not two consecutive nonnegative integers, e.g., 6 = 1 + 2 + 3, 12 = 3 + 4 + 5, 18 = 5 + 6 + 7, etc. (see A138591). - _Martin Renner_, Mar 14 2016 (Corrected by _David A. Corneth_, Aug 12 2016)
%C Numbers with three consecutive divisors: for some k, each of k, k+1, and k+2 divide n. - _Charles R Greathouse IV_, May 16 2016
%C Numbers k for which {phi(k),phi(2k),phi(3k)} is an arithmetic progression. - _Ivan Neretin_, Aug 12 2016
%H Vincenzo Librandi, <a href="/A008588/b008588.txt">Table of n, a(n) for n = 0..1000</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=318">Encyclopedia of Combinatorial Structures 318</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F From _Vincenzo Librandi_, Dec 24 2010: (Start)
%F a(n) = 6*n = 2*a(n-1) - a(n-2).
%F G.f.: 6*x/(1-x)^2. (End)
%F a(n) = Sum_{k>=0} A030308(n,k)*6*2^k. - _Philippe Deléham_, Oct 24 2011
%F a(n) = Sum_{k=2n-1..2n+1} k. - _Wesley Ivan Hurt_, Nov 22 2015
%F From _Ilya Gutkovskiy_, Aug 12 2016: (Start)
%F E.g.f.: 6*x*exp(x).
%F Convolution of A010722 and A057427.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/6 = A002162*A020793. (End)
%F a(n) = 6 * A001477(n). - _David A. Corneth_, Aug 12 2016
%p [ seq(6*n,n=0..45) ];
%t Range[0, 500, 6] (* _Vladimir Joseph Stephan Orlovsky_, May 26 2011 *)
%o (Magma) [6*n: n in [0..60] ]; // _Vincenzo Librandi_, Jul 16 2011
%o (PARI) a(n)=6*n \\ _Charles R Greathouse IV_, Feb 08 2012
%o (Maxima) makelist(6*n,n,0,30); /* _Martin Ettl_, Nov 12 2012 */
%o (Haskell)
%o a008588 = (* 6)
%o a008588_list = [0, 6 ..] -- _Reinhard Zumkeller_, Nov 10 2013
%Y Essentially the same as A008458.
%Y Cf. A016921, A016933, A016945, A016957, A016969, A138591.
%Y Cf. A044102 (subsequence).
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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