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A000351 Powers of 5: a(n) = 5^n.

%I A000351 M3937 N1620 #180 Mar 14 2025 20:01:58
%S A000351 1,5,25,125,625,3125,15625,78125,390625,1953125,9765625,48828125,
%T A000351 244140625,1220703125,6103515625,30517578125,152587890625,
%U A000351 762939453125,3814697265625,19073486328125,95367431640625,476837158203125,2384185791015625,11920928955078125
%N A000351 Powers of 5: a(n) = 5^n.
%C A000351 Same as Pisot sequences E(1, 5), L(1, 5), P(1, 5), T(1, 5). Essentially same as Pisot sequences E(5, 25), L(5, 25), P(5, 25), T(5, 25). See A008776 for definitions of Pisot sequences.
%C A000351 a(n) has leading digit 1 if and only if n = A067497 - 1. - _Lekraj Beedassy_, Jul 09 2002
%C A000351 With interpolated zeros 0, 1, 0, 5, 0, 25, ... (g.f.: x/(1 - 5*x^2)) second inverse binomial transform of Fibonacci(3n)/Fibonacci(3) (A001076). Binomial transform is A085449. - _Paul Barry_, Mar 14 2004
%C A000351 Sums of rows of the triangles in A013620 and A038220. - _Reinhard Zumkeller_, May 14 2006
%C A000351 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 less than 5. a(2) = 25 there are 25 compositions of natural numbers into 2 parts less than 5. - _Adi Dani_, Jun 22 2011
%C A000351 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 A000351 Numbers n such that sigma(5n) = 5n + sigma(n). In fact we have this theorem: p is a prime if and only if all solutions of the equation sigma(p*x) = p*x + sigma(x) are powers of p. - _Jahangeer Kholdi_, Nov 23 2013
%C A000351 From _Doug Bell_, Jun 22 2015: (Start)
%C A000351 Empirical observation: Where n is an odd multiple of 3, let x = (a(n) + 1)/9 and let y be the decimal expansion of x/a(n); then y*(x+1)/x + 1 = y rotated to the left.
%C A000351 Example:
%C A000351   a(3) = 125;
%C A000351   x = (125 + 1)/9 = 14;
%C A000351   y = 112, which is the decimal expansion of 14/125 = 0.112;
%C A000351   112*(14 + 1)/14 + 1 = 121 = 112 rotated to the left.
%C A000351 (End)
%C A000351 a(n) is the number of n-digit integers that contain only odd digits (A014261). - _Bernard Schott_, Nov 12 2022
%C A000351 Number of pyramids in the Sierpinski fractal square-based pyramid at the n-th step, while A279511 gives the corresponding number of vertices (see IREM link with drawings). - _Bernard Schott_, Nov 29 2022
%D A000351 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000351 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000351 T. D. Noe, <a href="/A000351/b000351.txt">Table of n, a(n) for n=0..100</a>
%H A000351 O. M. Cain, <a href="https://arxiv.org/abs/1910.13829">The Exceptional Selfcondensability of Powers of Five</a>, arXiv:1910.13829 [math.HO], 2019.
%H A000351 Peter 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 A000351 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=270">Encyclopedia of Combinatorial Structures 270</a>
%H A000351 IREM Paris-Nord, <a href="http://www-irem.univ-paris13.fr/site_spip/spip.php?article369">La pyramide de Sierpinski</a> (in French).
%H A000351 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A000351 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 A000351 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A000351 Yash Puri and Thomas 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 A000351 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BoxFractal.html">Box Fractal</a>
%H A000351 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (5).
%F A000351 a(n) = 5^n.
%F A000351 a(0) = 1; a(n) = 5*a(n-1) for n > 0.
%F A000351 G.f.: 1/(1 - 5*x).
%F A000351 E.g.f.: exp(5*x).
%F A000351 a(n) = A006495(n)^2 + A006496(n)^2.
%F A000351 a(n) = A159991(n) / A001021(n). - _Reinhard Zumkeller_, May 02 2009
%F A000351 From _Bernard Schott_, Nov 12 2022: (Start)
%F A000351 Sum_{n>=0} 1/a(n) = 5/4.
%F A000351 Sum_{n>=0} (-1)^n/a(n) = 5/6. (End)
%F A000351 a(n) = Sum_{k=0..n} C(2*n+1,n-k)*A000045(2*k+1). - _Vladimir Kruchinin_, Jan 14 2025
%p A000351 [ seq(5^n,n=0..30) ];
%p A000351 A000351:=-1/(-1+5*z); # _Simon Plouffe_ in his 1992 dissertation
%t A000351 Table[5^n, {n, 0, 30}] (* _Stefan Steinerberger_, Apr 06 2006 *)
%t A000351 5^Range[0, 30] (* _Harvey P. Dale_, Aug 22 2011 *)
%o A000351 (PARI) a(n)=5^n \\ _Charles R Greathouse IV_, Jun 10 2011
%o A000351 (Haskell)
%o A000351 a000351 = (5 ^)
%o A000351 a000351_list = iterate (* 5) 1  -- _Reinhard Zumkeller_, Oct 31 2012
%o A000351 (Maxima) makelist(5^n,n,0,20); /* _Martin Ettl_, Dec 27 2012 */
%o A000351 (Magma) [5^n : n in [0..30]]; // _Wesley Ivan Hurt_, Sep 27 2016
%o A000351 (Scala) (List.fill(50)(5: BigInt)).scanLeft(1: BigInt)(_ * _) // _Alonso del Arte_, May 31 2019
%o A000351 (Python)
%o A000351 def a(n): return 5**n
%o A000351 print([a(n) for n in range(24)]) # _Michael S. Branicky_, Nov 12 2022
%Y A000351 Cf. A009969 (even bisection), A013710 (odd bisection), A005054 (first differences), A003463 (partial sums).
%Y A000351 Cf. A006495, A006496, A159991, A001021.
%Y A000351 Cf. A001076, A008776, A013620, A038220, A067497, A085449, A014261.
%Y A000351 Sierpinski fractal square-based pyramid: A020858 (Hausdorff dimension), A279511 (number of vertices), this sequence (number of pyramids).
%K A000351 easy,nonn,nice
%O A000351 0,2
%A A000351 _N. J. A. Sloane_