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%I M4555 N1937 #111 Jul 12 2023 12:21:17
%S 1,8,64,512,4096,32768,262144,2097152,16777216,134217728,1073741824,
%T 8589934592,68719476736,549755813888,4398046511104,35184372088832,
%U 281474976710656,2251799813685248,18014398509481984,144115188075855872,1152921504606846976,9223372036854775808,73786976294838206464,590295810358705651712,4722366482869645213696
%N Powers of 8: a(n) = 8^n.
%C Same as Pisot sequences E(1, 8), L(1, 8), P(1, 8), T(1, 8). Essentially same as Pisot sequences E(8, 64), L(8, 64), P(8, 64), T(8, 64). See A008776 for definitions of Pisot sequences.
%C If X_1, X_2, ..., X_n is a partition of the set {1..2n} into blocks of size 2 then, for n>=1, a(n) is equal to the number of functions f : {1..2n} -> {1,2,3} such that for fixed y_1,y_2,...,y_n in {1,2,3} we have f(X_i)<>{y_i}, (i=1..n). - _Milan Janjic_, May 24 2007
%C This is the auto-convolution (convolution square) of A059304. - _R. J. Mathar_, May 25 2009
%C The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n>=1, a(n) equals the number of 8-colored compositions of n such that no adjacent parts have the same color. - _Milan Janjic_, Nov 17 2011
%C a(n) is equal to the determinant of a 3 X 3 matrix with rows 2^(n+2), 2^(n+1), 2^n; 2^(n+3), 2^(n+4), 2(n+3); 2^n, 2^(n+1), 2^(n+2) when it is divided by 144. - _J. M. Bergot_, May 07 2014
%C a(n) gives the number of small squares in the n-th iteration of the Sierpinski carpet fractal. Equivalently, the number of vertices in the n-Sierpinski carpet graph. - _Allan Bickle_, Nov 27 2022
%D K. H. Rosen et al., eds., Handbook of Discrete and Combinatorial Mathematics, CRC Press, 2017; p. 15.
%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="/A001018/b001018.txt">Table of n, a(n) for n = 0..100</a>
%H Allan Bickle, <a href="https://allanbickle.files.wordpress.com/2016/05/mengerspongedegree.pdf">Degrees of Menger and Sierpinski Graphs</a>, Congr. Num. 227 (2016) 197-208.
%H Allan Bickle, <a href="https://allanbickle.files.wordpress.com/2016/05/mengerspongeshort.pdf">MegaMenger Graphs</a>, The College Mathematics Journal, 49 1 (2018) 20-26.
%H P. J. Cameron, <a href="http://www.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=273">Encyclopedia of Combinatorial Structures 273</a>
%H Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Enumerative Formulas for Some Functions on Finite Sets</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H Caroline Nunn, <a href="https://scholar.rose-hulman.edu/rhumj/vol22/iss2/3">A Proof of a Generalization of Niven's Theorem Using Algebraic Number Theory</a>, Rose-Hulman Undergraduate Mathematics Journal: Vol. 22, Iss. 2, Article 3 (2021). See table at p. 9.
%H Y. Puri and T. Ward, <a href="http://www.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/SierpinskiCarpet.html">Sierpiński Carpet</a>
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (8).
%F a(n) = 8^n.
%F a(0) = 1; a(n) = 8*a(n-1) for n > 0.
%F G.f.: 1/(1-8*x).
%F E.g.f.: exp(8*x).
%F Sum_{n>=0} 1/a(n) = 8/7. - _Gary W. Adamson_, Aug 29 2008
%F a(n) = A157176(A008588(n)); a(n+1) = A157176(A016969(n)). - _Reinhard Zumkeller_, Feb 24 2009
%F From _Stefano Spezia_, Dec 28 2021: (Start)
%F a(n) = (-1)^n*(1 + sqrt(-3))^(3*n) (see Nunn, p. 9).
%F a(n) = (-1)^n*Sum_{k=0..floor(3*n/2)} (-3)^k*binomial(3*n, 2*k) (see Nunn, p. 9). (End)
%e For n=1, the 1st order Sierpinski carpet graph is an 8-cycle.
%p seq(8^n, n=0..23); # _Nathaniel Johnston_, Jun 26 2011
%p A001018 := n -> 8^n; # _M. F. Hasler_, Apr 19 2015
%t Table[8^n, {n,0,50}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 15 2011 *)
%o (Maxima) makelist(8^n,n,0,20); /* _Martin Ettl_, Nov 12 2012 */
%o (PARI) a(n)=8^n \\ _Charles R Greathouse IV_, May 10 2014
%o (Haskell)
%o a001018 = (8 ^)
%o a001018_list = iterate (* 8) 1 -- _Reinhard Zumkeller_, Apr 29 2015
%o (Magma) [8^n : n in [0..30]]; // _Wesley Ivan Hurt_, Sep 27 2016
%o (Python)
%o print([8**n for n in range(25)]) # _Michael S. Branicky_, Dec 29 2021
%Y Cf. A013730, A103333, A013731, A067417, A083233, A055274.
%Y Cf. A008588, A016969, A157176.
%Y Cf. A000079 (powers of 2), A000244 (powers of 3), A000302 (powers of 4), A000351 (powers of 5), A000400 (powers of 6), A000420 (powers of 7), A001019 (powers of 9), ..., A001029 (powers of 19), A009964 (powers of 20), ..., A009992 (powers of 48), A087752 (powers of 49), A165800 (powers of 50), A159991 (powers of 60).
%Y Cf. A032766 (floor(3*n/2)).
%Y Cf. A271939 (number of edges in the n-Sierpinski carpet graph).
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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