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A005054 a(n) = (4*5^n + 0^n) / 5. 15
1, 4, 20, 100, 500, 2500, 12500, 62500, 312500, 1562500, 7812500, 39062500, 195312500, 976562500, 4882812500, 24414062500, 122070312500, 610351562500, 3051757812500, 15258789062500, 76293945312500 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

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

Eric Weisstein's World of Mathematics, Fermat Number

FORMULA

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

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

a(0) = 1, a(1) = 4, 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

MATHEMATICA

f[n_]:=(4*5^n)/5; lst={}; Do[AppendTo[lst, f[n]], {n, 4!}]; lst

a=1; s=a; lst={a}; Do[AppendTo[lst, a=4*s]; s=a+s, {n, 4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Nov 10 2009 *)

Table[EulerPhi[5^n], {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Nov 10 2009 *)

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

PROG

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

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

CROSSREFS

First differences of A000351. Cf. A000215.

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

Sequence in context: A073532 A178874 A103771 * A216099 A105480 A242156

Adjacent sequences:  A005051 A005052 A005053 * A005055 A005056 A005057

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

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Last modified March 22 22:17 EDT 2017. Contains 283901 sequences.