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

1, 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, 9765625, 48828125, 244140625, 1220703125, 6103515625, 30517578125, 152587890625, 762939453125, 3814697265625, 19073486328125, 95367431640625, 476837158203125, 2384185791015625, 11920928955078125

Offset: 0

Author

N. J. A. Sloane

Keywords

• easy
• nonn
• nice

Comments

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.

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

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

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 less than 5. a(2) = 25 there are 25 compositions of natural numbers into 2 parts less than 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 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

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 be the 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)

a(n) is the number of n-digit integers that contain only odd digits (A014261). - Bernard Schott, Nov 12 2022

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

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
• O. M. Cain, The Exceptional Selfcondensability of Powers of Five, arXiv:1910.13829 [math.HO], 2019.
• Peter 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
• IREM Paris-Nord, La pyramide de Sierpinski (in French).
• Tanya Khovanova, Recursive Sequences
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• Yash Puri and Thomas 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).

Crossrefs

Cf. A009969 (even bisection), A013710 (odd bisection), A005054 (first differences), A003463 (partial sums).

Cf. A006495, A006496, A159991, A001021.

Cf. A001076, A008776, A013620, A038220, A067497, A085449, A014261.

Sierpinski fractal square-based pyramid: A020858 (Hausdorff dimension), A279511 (number of vertices), this sequence (number of pyramids).

Programs

Haskell

a000351 = (5 ^)
a000351_list = iterate (* 5) 1  -- Reinhard Zumkeller, Oct 31 2012

Magma

[5^n : n in [0..30]]; // Wesley Ivan Hurt, Sep 27 2016

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 *)

Maxima

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

PARI

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

Python

def a(n): return 5**n
print([a(n) for n in range(24)]) # Michael S. Branicky, Nov 12 2022

Scala

(List.fill(50)(5: BigInt)).scanLeft(1: BigInt)( * ) // Alonso del Arte, May 31 2019

Formula

a(n) = 5^n.

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

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

From Bernard Schott, Nov 12 2022: (Start)

Sum_{n>=0} 1/a(n) = 5/4.

Sum_{n>=0} (-1)^n/a(n) = 5/6. (End)

a(n) = Sum_{k=0..n} C(2*n+1,n-k)*A000045(2*k+1). - Vladimir Kruchinin, Jan 14 2025