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%I M4914 N2107 #76 Jul 12 2023 12:26:38
%S 1,13,169,2197,28561,371293,4826809,62748517,815730721,10604499373,
%T 137858491849,1792160394037,23298085122481,302875106592253,
%U 3937376385699289,51185893014090757,665416609183179841,8650415919381337933,112455406951957393129,1461920290375446110677,19004963774880799438801
%N Powers of 13.
%C Same as Pisot sequences E(1, 13), L(1, 13), P(1, 13), T(1, 13). Essentially same as Pisot sequences E(13, 169), L(13, 169), P(13, 169), T(13, 169). See A008776 for definitions of Pisot sequences.
%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 13-colored compositions of n such that no adjacent parts have the same color. - _Milan Janjic_, Nov 17 2011
%C Numbers n such that sigma(13*n) = 13*n+sigma(n). - _Jahangeer Kholdi_, Nov 23 2013
%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="/A001022/b001022.txt">Table of n, a(n) for n = 0..100</a>
%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=277">Encyclopedia of Combinatorial Structures 277</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.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%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 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (13).
%F G.f.: 1/(1-13*x).
%F E.g.f.: exp(13*x).
%F a(n) = 13^n. - _Vincenzo Librandi_, Nov 21 2010
%F a(n) = 13*a(n-1) n > 0, a(0)=1. - _Vincenzo Librandi_, Nov 21 2010
%F a(n) = Sum_{k=0..n} A001021(k)*binomial(n,k). It is well known that Sum_{k=0..n} (h-1)^k*binomial(n,k) = h^n. - _Bruno Berselli_, Aug 06 2013
%e For the fifth formula: a(7) = 1*1 + 12*7 + 144*21 + 1728*35 + 20736*35 + 248832*21 + 2985984*7 + 35831808*1 = 62748517. - _Bruno Berselli_, Aug 06 2013
%p A001022:=-1/(-1+13*z); # _Simon Plouffe_ in his 1992 dissertation
%t Table[13^n, {n, 0, 40}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 15 2011 *)
%o (Magma) [13^n: n in [0..100]]; // _Vincenzo Librandi_, Nov 21 2010
%o (Maxima) A001022(n):=13^n$ makelist(A001022(n),n,0,30); /* _Martin Ettl_, Nov 05 2012 */
%o (PARI) first(n)=powers(13,n) \\ _Charles R Greathouse IV_, Jun 17 2021
%Y Cf. A001021.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _James A. Sellers_, Sep 19 2000
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