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A000351 Powers of 5.
(Formerly M3937 N1620)
163
1, 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, 9765625, 48828125, 244140625, 1220703125, 6103515625, 30517578125, 152587890625, 762939453125, 3814697265625, 19073486328125, 95367431640625, 476837158203125, 2384185791015625, 11920928955078125 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

a(n) has leading digit 1 iff n = A067497 - 1. - Lekraj Beedassy, Jul 09 2002

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

Sums of rows of the triangles in A013620 and A038220. - Reinhard Zumkeller, May 14 2006

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

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

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

From Doug Bell, Jun 22 2015: (Start)

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.

Example:

  a(3) = 125;

  x = (125+1)/9 = 14;

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

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

(End)

REFERENCES

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n=0..100

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 270

Tanya Khovanova, Recursive Sequences

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.

Eric Weisstein's World of Mathematics, Box Fractal

Index entries for linear recurrences with constant coefficients, signature (5).

FORMULA

a(n) = 5^n.

a(0) = 1; a(n) = 5*a(n-1).

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

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

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

a(n) = A159991(n)/A001021(n). - Reinhard Zumkeller, May 02 2009

MAPLE

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

A000351:=-1/(-1+5*z); # Simon Plouffe in his 1992 dissertation

MATHEMATICA

Table[5^n, {n, 0, 30}] (* Stefan Steinerberger, Apr 06 2006 *)

5^Range[0, 30] (* Harvey P. Dale, Aug 22 2011 *)

PROG

(PARI) a(n)=5^n \\ Charles R Greathouse IV, Jun 10 2011

(Haskell)

a000351 = (5 ^)

a000351_list = iterate (* 5) 1  -- Reinhard Zumkeller, Oct 31 2012

(Maxima) makelist(5^n, n, 0, 20); /* Martin Ettl, Dec 27 2012 */

CROSSREFS

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

a(n) = A159991(n)/A001021(n). - Reinhard Zumkeller, May 02 2009

Sequence in context: A129066 A102169 A060391 * A050735 A195948 A083590

Adjacent sequences:  A000348 A000349 A000350 * A000352 A000353 A000354

KEYWORD

easy,nonn,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

Removed attribute "conjectured" from Simon Plouffe g.f., R. J. Mathar, Mar 11 2009

STATUS

approved

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Last modified May 26 06:46 EDT 2016. Contains 273322 sequences.