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%I A000400 M4224 N1765 #147 Jul 07 2024 13:46:25
%S A000400 1,6,36,216,1296,7776,46656,279936,1679616,10077696,60466176,
%T A000400 362797056,2176782336,13060694016,78364164096,470184984576,
%U A000400 2821109907456,16926659444736,101559956668416,609359740010496,3656158440062976,21936950640377856,131621703842267136
%N A000400 Powers of 6: a(n) = 6^n.
%C A000400 Same as Pisot sequences E(1, 6), L(1, 6), P(1, 6), T(1, 6). Essentially same as Pisot sequences E(6, 36), L(6, 36), P(6, 36), T(6, 36). See A008776 for definitions of Pisot sequences.
%C A000400 Central terms of the triangle in A036561. - _Reinhard Zumkeller_, May 14 2006
%C A000400 a(n) = A169604(n)/3; a(n+1) = 2*A169604(n). - _Reinhard Zumkeller_, May 02 2010
%C A000400 Number of pentagons contained within pentaflakes. - _William A. Tedeschi_, Sep 12 2010
%C A000400 Sum of coefficients of expansion of (1 + x + x^2 + x^3 + x^4 + x^5)^n.
%C A000400 a(n) is number of compositions of natural numbers into n parts less than 6. For example, a(2) = 36, and there are 36 compositions of natural numbers into 2 parts less than 6.
%C A000400 The compositions of n in which each part is colored by one of  p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 5-colored compositions of n such that no adjacent parts have the same color.
%C A000400 Number of words of length n over the alphabet of six letters. - _Joerg Arndt_, Sep 16 2014
%C A000400 The number of ordered triples (x, y, z) of binary words of length n such that D(x,z) = D(x, y) + D(y, z) where D(a, b) is the Hamming distance from a to b. - _Geoffrey Critzer_, Mar 06 2017
%C A000400 a(n) is the area of a triangle with vertices at (2^n, 3^n), (2^(n+1), 3^(n+1)), and (2^(n+2), 3^(n+2)); a(n) is also one fifth the area of a triangle with vertices at (2^n, 3^(n+2)), (2^(n+1), 3^(n+1)), and (2^(n+2), 3^n). - _J. M. Bergot_, May 07 2018
%C A000400 a(n) is the number of possible outcomes of n distinguishable 6-sided dice. - _Stefano Spezia_, Jul 06 2024
%D A000400 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000400 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000400 T. D. Noe, <a href="/A000400/b000400.txt">Table of n, a(n) for n = 0..100</a>
%H A000400 C. Banderier and D. Merlini, <a href="http://algo.inria.fr/banderier/Papers/infjumps.ps">Lattice paths with an infinite set of jumps</a>, FPSAC02, Melbourne, 2002.
%H A000400 Peter J. Cameron, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000400 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=271">Encyclopedia of Combinatorial Structures 271</a>
%H A000400 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A000400 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A000400 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A000400 Yash Puri and Thomas Ward, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL4/WARD/short.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%H A000400 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Pentaflake.html">Pentaflake</a>
%H A000400 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (6).
%F A000400 a(n) = 6^n.
%F A000400 a(0) = 1; a(n) = 6*a(n-1).
%F A000400 G.f.: 1/(1-6*x). - _Simon Plouffe_ in his 1992 dissertation.
%F A000400 E.g.f.: exp(6*x).
%F A000400 A000005(a(n)) = A000290(n+1). - _Reinhard Zumkeller_, Mar 04 2007
%F A000400 a(n) = A159991(n)/A011577(n). - _Reinhard Zumkeller_, May 02 2009
%F A000400 a(n) = det(|s(i+3,j)|, 1 <= i,j <= n), where s(n,k) are Stirling numbers of the first kind. - _Mircea Merca_, Apr 04 2013
%t A000400 6^Range[0, 40] (* _Harvey P. Dale_, Jan 24 2013 *)
%o A000400 (PARI) a(n)=6^n \\ _Charles R Greathouse IV_, Jun 16 2011
%o A000400 (Maxima) A000400(n):=6^n$
%o A000400 makelist(A000400(n),n,0,30); /* _Martin Ettl_, Nov 05 2012 */
%o A000400 (Haskell)
%o A000400 a000400 = (6 ^)
%o A000400 a000400_list = iterate (* 6) 1  -- _Reinhard Zumkeller_, Nov 21 2013
%o A000400 (Scala) (List.fill(50)(6: BigInt)).scanLeft(1: BigInt)(_ * _) // _Alonso del Arte_, May 31 2019
%Y A000400 Column 3 of A225816.
%Y A000400 Row 6 of A003992.
%Y A000400 Row 3 of A329332.
%Y A000400 Cf. A000005, A000290, A011577, A036561, A159991, A169604,
%K A000400 easy,nonn
%O A000400 0,2
%A A000400 _N. J. A. Sloane_

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