A005054 a(0) = 1; a(n) = 4*5^(n-1) for n >= 1.

1, 4, 20, 100, 500, 2500, 12500, 62500, 312500, 1562500, 7812500, 39062500, 195312500, 976562500, 4882812500, 24414062500, 122070312500, 610351562500, 3051757812500, 15258789062500, 76293945312500, 381469726562500, 1907348632812500, 9536743164062500

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Consider the sequence formed by the final n decimal digits of {2^k: k >= 0}. For n=1 this is 1, 2, 4, 8, 6, 2, 4, ... (A000689) with period 4. For any n this is periodic with period a(n). Cf. A000855 (n=2), A126605 (n=3, also n=4). - N. J. A. Sloane, Jul 08 2022

First differences of A000351.

Length of repeating cycle of the final n+1 digits in Fermat numbers. - Lekraj Beedassy, Robert G. Wilson v and Eric W. Weisstein, Jul 05 2004

Number of n-digit endings for a power of 2 whose exponent is greater than or equal to n. - J. Lowell

For n>=1, a(n) is equal to the number of functions f:{1,2,...,n}->{1,2,3,4,5} such that for a fixed x in {1,2,...,n} and a fixed y in {1,2,3,4,5} we have f(x) != y. - Aleksandar M. Janjic and Milan Janjic, Mar 27 2007

Equals INVERT transform of A033887: (1, 3, 13, 55, 233, ...) and INVERTi transform of A001653: (1, 5, 29, 169, 985, 5741, ...). - Gary W. Adamson, Jul 22 2010

a(n) = (n+1) terms in the sequence (1, 3, 4, 4, 4, ...) dot (n+1) terms in the sequence (1, 1, 4, 20, 100, ...). Example: a(4) = 500 = (1, 3, 4, 4, 4) dot (1, 1, 4, 20, 100) = (1 + 3 + 16, + 80 + 400), where (1, 3, 16, 80, 400, ...) = A055842, finite differences of A005054 terms. - Gary W. Adamson, Aug 03 2010

a(n) is the number of compositions of n when there are 4 types of each natural number. - Milan Janjic, Aug 13 2010

Apart from the first term, number of monic squarefree polynomials over F_5 of degree n. - Charles R Greathouse IV, Feb 07 2012

For positive integers that can be either of two colors (designated by ' or ''), a(n) is the number of compositions of 2n that are cardinal palindromes; that is, palindromes that only take into account the cardinality of the numbers and not their colors. Example: 3', 2'', 1', 1, 2', 3'' would count as a cardinal palindrome. - Gregory L. Simay, Mar 01 2020

a(n) is the length of the period of the sequence Fibonacci(k) (mod 5^(n-1)) (for n>1) and the length of the period of the sequence Lucas(k) (mod 5^n) (Kramer and Hoggatt, 1972). - Amiram Eldar, Feb 02 2022

References

• T. Koshy, "The Ends Of A Fermat Number", pp. 183-4 Journal Recreational Mathematics, vol. 31(3) 2002-3 Baywood NY.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 458.
• Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
• Judy Kramer and V. E. Hoggatt, Jr., Special Cases of Fibonacci Periodicity (Part 1, Part 2), The Fibonacci Quarterly, Vol. 10, No. 5 (1972), pp. 519-522, 530.
• Eric Weisstein's World of Mathematics, Fermat Number.
• Index entries for linear recurrences with constant coefficients, signature (5).

Crossrefs

Cf. A000010, A000032, A000045, A000351, A000215, A055842.

Cf. A001653, A033887. - Gary W. Adamson, Jul 22 2010

See also A000689, A000855, A126605, A216099, A216164. - N. J. A. Sloane, Jul 08 2022

Programs

Magma

[(4*5^n+0^n)/5: n in [0..30]]; // Vincenzo Librandi, Jun 08 2013

Maple

a:= n-> ceil(4*5^(n-1)):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 08 2022

Mathematica

CoefficientList[Series[(1 - x) / (1 - 5 x), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 08 2013 *)

PARI

Vec((1-x)/(1-5*x) + O(x^100)) \\ Altug Alkan, Dec 07 2015

Formula

a(n) = (4*5^n + 0^n) / 5. - R. J. Mathar, May 13 2008

G.f.: (1-x)/(1-5*x). - Philippe Deléham, Nov 02 2009

G.f.: 1/(1 - 4*Sum_{k>=1} x^k).

a(n) = 5*a(n-1) for n>=2. - Vincenzo Librandi, Dec 31 2010

a(n) = phi(5^n) = A000010(A000351(n)).

E.g.f.: (4*exp(5*x)+1)/5. - Paul Barry, Apr 20 2003

a(n + 1) = (((1 + sqrt(-19))/2)^n + ((1 - sqrt(-19))/2)^n)^2 - (((1 + sqrt(-19))/2)^n - ((1 - sqrt(-19))/2)^n)^2. - Raphie Frank, Dec 07 2015

Extensions

Better definition from R. J. Mathar, May 13 2008

Edited by N. J. A. Sloane, Jul 08 2022