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 A000351 Powers of 5: a(n) = 5^n. (Formerly M3937 N1620) 177

%I M3937 N1620

%S 1,5,25,125,625,3125,15625,78125,390625,1953125,9765625,48828125,

%T 244140625,1220703125,6103515625,30517578125,152587890625,

%U 762939453125,3814697265625,19073486328125,95367431640625,476837158203125,2384185791015625,11920928955078125

%N Powers of 5: a(n) = 5^n.

%C Same as Pisot sequences E(1,5), L(1,5), P(1,5), T(1,5). See A008776 for definitions of Pisot sequences.

%C a(n) has leading digit 1 iff n = A067497 - 1. - _Lekraj Beedassy_, Jul 09 2002

%C With interpolated zeros 0,1,0,5,0,25,... (G.f.: x/(1-5*x^2)) second inverse binomial transform of Fib(3n)/F(3) (A001076). Binomial transform is A085449. - _Paul Barry_, Mar 14 2004

%C Sums of rows of the triangles in A013620 and A038220. - _Reinhard Zumkeller_, May 14 2006

%C Sum of coefficients of expansion of (1+x+x^2+x^3+x^4)^n. a(n) is number of compositions of natural numbers into n parts < 5. a(2)=25 there are 25 compositions of natural numbers into 2 parts < 5. - _Adi Dani_, Jun 22 2011

%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 5-colored compositions of n such that no adjacent parts have the same color. - _Milan Janjic_, Nov 17 2011

%C Numbers n such that sigma(5n) = 5n + sigma(n). In fact we have this theorem: p is a prime iff all solutions of the equation sigma(p*x) = p*x + sigma(x) are powers of p. - _Jahangeer Kholdi_, Nov 23 2013

%C From _Doug Bell_, Jun 22 2015: (Start)

%C Empirical observation: Where n is an odd multiple of 3, let x = (a(n)+1)/9 and let y = decimal expansion of x/a(n); then y*(x+1)/x + 1 = y rotated to the left.

%C Example:

%C a(3) = 125;

%C x = (125+1)/9 = 14;

%C y = 112, which is the decimal expansion of 14/125 = 0.112;

%C 112*(14+1)/14 + 1 = 121 = 112 rotated to the left.

%C (End)

%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="/A000351/b000351.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=270">Encyclopedia of Combinatorial Structures 270</a>

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992.

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Université du Québec à Montréal, 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BoxFractal.html">Box Fractal</a>

%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (5).

%F a(n) = 5^n.

%F a(0) = 1; a(n) = 5*a(n-1) for n > 0.

%F G.f.: 1/(1-5*x).

%F E.g.f.: exp(5*x).

%F a(n) = A006495(n)^2 + A006496(n)^2.

%F a(n) = A159991(n) / A001021(n). - _Reinhard Zumkeller_, May 02 2009

%p [ seq(5^n,n=0..30) ];

%p A000351:=-1/(-1+5*z); # _Simon Plouffe_ in his 1992 dissertation

%t Table[5^n, {n, 0, 30}] (* _Stefan Steinerberger_, Apr 06 2006 *)

%t 5^Range[0,30] (* _Harvey P. Dale_, Aug 22 2011 *)

%o (PARI) a(n)=5^n \\ _Charles R Greathouse IV_, Jun 10 2011

%o a000351 = (5 ^)

%o a000351_list = iterate (* 5) 1 -- _Reinhard Zumkeller_, Oct 31 2012

%o (Maxima) makelist(5^n,n,0,20); /* _Martin Ettl_, Dec 27 2012 */

%o (MAGMA) [5^n : n in [0..30]]; // _Wesley Ivan Hurt_, Sep 27 2016

%Y a(n) = A006495(n)^2 + A006496(n)^2.

%Y a(n) = A159991(n) / A001021(n). - _Reinhard Zumkeller_, May 02 2009

%Y Cf. A001076, A008776, A013620, A038220, A067497, A085449.

%K easy,nonn,nice

%O 0,2

%A _N. J. A. Sloane_

%E Removed attribute "conjectured" from _Simon Plouffe_ g.f., _R. J. Mathar_, Mar 11 2009

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