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%I M4949 N2120 #83 Jul 12 2023 12:27:13
%S 1,14,196,2744,38416,537824,7529536,105413504,1475789056,20661046784,
%T 289254654976,4049565169664,56693912375296,793714773254144,
%U 11112006825558016,155568095557812224,2177953337809371136,30491346729331195904,426878854210636742656,5976303958948914397184,83668255425284801560576
%N Powers of 14.
%C Same as Pisot sequences E(1, 14), L(1, 14), P(1, 14), T(1, 14). Essentially same as Pisot sequences E(14, 196), L(14, 196), P(14, 196), T(14, 196). See A008776 for definitions of Pisot sequences.
%C Number of n-permutations of 15 objects: l, m, n, o, p, q, r, s, t, u, v, w, z, x, y with repetition allowed and containing no u's, (u-free). Permutations with repetitions! If n=0 then 1 >>14^0=1 "". (no u's.) If n=1 then 13 >>14^1=14, >> l, m, n, o, p, q, r, s, t, v, w, z, x, y. (no u's.) etc. - _Zerinvary Lajos_, Jul 01 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 14-colored compositions of n such that no adjacent parts have the same color. - _Milan Janjic_, Nov 17 2011
%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="/A001023/b001023.txt">Table of n, a(n) for n = 0..100</a>
%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=278">Encyclopedia of Combinatorial Structures 278</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 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (14).
%F G.f.: 1/(1-14x), e.g.f.: exp(14x)
%F A000005(a(n)) = A000290(n+1). - _Reinhard Zumkeller_, Mar 04 2007
%F a(n) = 14^n; a(n) = 14*a(n-1) with a(0)=1. - _Vincenzo Librandi_, Nov 21 2010
%p A001023:=-1/(-1+14*z); # conjectured by _Simon Plouffe_ in his 1992 dissertation
%t Table[14^n,{n,0,40}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 15 2011 *)
%t Denominator/@HermiteH[Range[0,20],5/28] (* _Harvey P. Dale_, Jul 11 2011 *)
%o (Sage) [lucas_number1(n,14,0) for n in range(1, 18)]# _Zerinvary Lajos_, Apr 29 2009
%o (Magma) [ 14^n: n in [0..20] ]; // _Vincenzo Librandi_, Nov 21 2010
%o (Magma) [ n eq 1 select 1 else 14*Self(n-1): n in [1..21] ];
%o (PARI) a(n)=14^n \\ _Charles R Greathouse IV_, Nov 18 2011
%o (Python) print([14**n for n in range(21)]) # _Michael S. Branicky_, Jan 14 2021
%Y Cf. A000005, A000290, A160193.
%Y Row 9 of A329332.
%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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