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%I #112 Apr 29 2023 08:09:18
%S 1,4,20,100,500,2500,12500,62500,312500,1562500,7812500,39062500,
%T 195312500,976562500,4882812500,24414062500,122070312500,610351562500,
%U 3051757812500,15258789062500,76293945312500,381469726562500,1907348632812500,9536743164062500
%N a(0) = 1; a(n) = 4*5^(n-1) for n >= 1.
%C 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
%C First differences of A000351.
%C 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
%C Number of n-digit endings for a power of 2 whose exponent is greater than or equal to n. - _J. Lowell_
%C 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
%C 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
%C 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
%C a(n) is the number of compositions of n when there are 4 types of each natural number. - _Milan Janjic_, Aug 13 2010
%C Apart from the first term, number of monic squarefree polynomials over F_5 of degree n. - _Charles R Greathouse IV_, Feb 07 2012
%C 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
%C 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
%D T. Koshy, "The Ends Of A Fermat Number", pp. 183-4 Journal Recreational Mathematics, vol. 31(3) 2002-3 Baywood NY.
%H Vincenzo Librandi, <a href="/A005054/b005054.txt">Table of n, a(n) for n = 0..1000</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=458">Encyclopedia of Combinatorial Structures 458</a>.
%H Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfun.pdf">Enumerative Formulas for Some Functions on Finite Sets</a>.
%H Judy Kramer and V. E. Hoggatt, Jr., Special Cases of Fibonacci Periodicity (<a href="https://www.fq.math.ca/Scanned/10-5/kramer-a.pdf">Part 1</a>, <a href="https://www.fq.math.ca/Scanned/10-5/kramer-b.pdf">Part 2</a>), The Fibonacci Quarterly, Vol. 10, No. 5 (1972), pp. 519-522, 530.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FermatNumber.html">Fermat Number</a>.
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (5).
%F a(n) = (4*5^n + 0^n) / 5. - _R. J. Mathar_, May 13 2008
%F G.f.: (1-x)/(1-5*x). - _Philippe Deléham_, Nov 02 2009
%F G.f.: 1/(1 - 4*Sum_{k>=1} x^k).
%F a(n) = 5*a(n-1) for n>=2. - _Vincenzo Librandi_, Dec 31 2010
%F a(n) = phi(5^n) = A000010(A000351(n)).
%F E.g.f.: (4*exp(5*x)+1)/5. - _Paul Barry_, Apr 20 2003
%F 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
%p a:= n-> ceil(4*5^(n-1)):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Jul 08 2022
%t CoefficientList[Series[(1 - x) / (1 - 5 x), {x, 0, 50}], x] (* _Vincenzo Librandi_, Jun 08 2013 *)
%o (Magma) [(4*5^n+0^n)/5: n in [0..30]]; // _Vincenzo Librandi_, Jun 08 2013
%o (PARI) Vec((1-x)/(1-5*x) + O(x^100)) \\ _Altug Alkan_, Dec 07 2015
%Y Cf. A000010, A000032, A000045, A000351, A000215, A055842.
%Y Cf. A001653, A033887. - _Gary W. Adamson_, Jul 22 2010
%Y See also A000689, A000855, A126605, A216099, A216164. - _N. J. A. Sloane_, Jul 08 2022
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E Better definition from _R. J. Mathar_, May 13 2008
%E Edited by _N. J. A. Sloane_, Jul 08 2022
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