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%I M4224 N1765 #147 Jul 07 2024 13:46:25
%S 1,6,36,216,1296,7776,46656,279936,1679616,10077696,60466176,
%T 362797056,2176782336,13060694016,78364164096,470184984576,
%U 2821109907456,16926659444736,101559956668416,609359740010496,3656158440062976,21936950640377856,131621703842267136
%N Powers of 6: a(n) = 6^n.
%C 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 Central terms of the triangle in A036561. - _Reinhard Zumkeller_, May 14 2006
%C a(n) = A169604(n)/3; a(n+1) = 2*A169604(n). - _Reinhard Zumkeller_, May 02 2010
%C Number of pentagons contained within pentaflakes. - _William A. Tedeschi_, Sep 12 2010
%C Sum of coefficients of expansion of (1 + x + x^2 + x^3 + x^4 + x^5)^n.
%C 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 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 Number of words of length n over the alphabet of six letters. - _Joerg Arndt_, Sep 16 2014
%C 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 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 a(n) is the number of possible outcomes of n distinguishable 6-sided dice. - _Stefano Spezia_, Jul 06 2024
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A000400/b000400.txt">Table of n, a(n) for n = 0..100</a>
%H 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 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 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=271">Encyclopedia of Combinatorial Structures 271</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H 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 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Pentaflake.html">Pentaflake</a>
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (6).
%F a(n) = 6^n.
%F a(0) = 1; a(n) = 6*a(n-1).
%F G.f.: 1/(1-6*x). - _Simon Plouffe_ in his 1992 dissertation.
%F E.g.f.: exp(6*x).
%F A000005(a(n)) = A000290(n+1). - _Reinhard Zumkeller_, Mar 04 2007
%F a(n) = A159991(n)/A011577(n). - _Reinhard Zumkeller_, May 02 2009
%F 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 6^Range[0, 40] (* _Harvey P. Dale_, Jan 24 2013 *)
%o (PARI) a(n)=6^n \\ _Charles R Greathouse IV_, Jun 16 2011
%o (Maxima) A000400(n):=6^n$
%o makelist(A000400(n),n,0,30); /* _Martin Ettl_, Nov 05 2012 */
%o (Haskell)
%o a000400 = (6 ^)
%o a000400_list = iterate (* 6) 1 -- _Reinhard Zumkeller_, Nov 21 2013
%o (Scala) (List.fill(50)(6: BigInt)).scanLeft(1: BigInt)(_ * _) // _Alonso del Arte_, May 31 2019
%Y Column 3 of A225816.
%Y Row 6 of A003992.
%Y Row 3 of A329332.
%Y Cf. A000005, A000290, A011577, A036561, A159991, A169604,
%K easy,nonn
%O 0,2
%A _N. J. A. Sloane_
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