A126605 -id:A126605 - cp's OEIS frontend

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

N. J. A. Sloane

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

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

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).

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

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

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

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

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

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

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

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

A000855 Final two digits of 2^n.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 28, 56, 12, 24, 48, 96, 92, 84, 68, 36, 72, 44, 88, 76, 52, 4, 8, 16, 32, 64, 28, 56, 12, 24, 48, 96, 92, 84, 68, 36, 72, 44, 88, 76, 52, 4, 8, 16, 32, 64, 28, 56, 12, 24, 48, 96, 92, 84, 68, 36, 72, 44, 88, 76, 52, 4, 8, 16, 32, 64, 28

Offset: 0

N. J. A. Sloane

Has period 20 (starting with a(2)=4).

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for sequences related to final digits of numbers
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,-1,1).

Cf. A000689, A126605.

PowerMod[2,Range[0,60],100] (* Harvey P. Dale, Jan 05 2012 *)

a(n)=lift(Mod(2,100)^n) \\ Charles R Greathouse IV, Dec 28 2012

G.f.: -(50*x^12 +25*x^11 +13*x^10 -44*x^9 +28*x^8 -36*x^7 +32*x^6 +16*x^5 +8*x^4 +4*x^3 +2*x^2 +x +1) / ((x -1)*(x^2 +1)*(x^8 -x^6 +x^4 -x^2 +1)). - Colin Barker, Dec 01 2014

For n > 13: a(n) = a(n-1) - a(n-10) + a(n-11). - Ray Chandler, Aug 09 2025

A181610 The last n digits of powers of 2 start cycling. a(n) is the number of zero-free terms in this cycle.

Original entry on oeis.org

4, 18, 81, 364, 1638, 7371, 33170, 149268, 671701, 3022653, 13601945, 61208743, 275439346, 1239477074, 5577646830, 25099410745, 112947348510, 508263067945, 2287183805359, 10292327123878, 46315472056678, 208419624257654, 937888309161430, 4220497391215744

Offset: 1

Tanya Khovanova, Jan 30 2011

			The last two digits of powers of two cycle through 20 terms (A000855): 04, 08, 16, 32, 64, 28, 56, 12, 24, 48, 96, 92, 84, 68, 36, 72, 44, 88, 76, 52. Out of those 18 do not contain a zero. Hence a(2) = 18.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..26

The corresponding cycle length is A005054. See A126605 for n=3.

f[n_] := Block[{c = 0, k = n, lmt = n + 4*5^(n - 1)},
While[k < lmt, m = PowerMod[2, k, 10^n];
  If[m >= 10^(n - 1) && !MemberQ[ IntegerDigits@ m, 0], c++ ]; k++ ]; c];
Array[ f, 11] (* Robert G. Wilson v, Jan 30 2011 *)

a(8)-a(11) from Robert G. Wilson v, Jan 30 2011

a(12)-a(24) from Hiroaki Yamanouchi, Mar 21 2015

A216095 a(n) = 2^n mod 10000.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 6384, 2768, 5536, 1072, 2144, 4288, 8576, 7152, 4304, 8608, 7216, 4432, 8864, 7728, 5456, 912, 1824, 3648, 7296, 4592, 9184, 8368, 6736, 3472, 6944, 3888, 7776, 5552, 1104, 2208, 4416, 8832, 7664, 5328, 656, 1312, 2624, 5248, 496, 992, 1984, 3968, 7936, 5872, 1744, 3488, 6976, 3952, 7904, 5808

Offset: 0

V. Raman, Sep 01 2012

Period = 500.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000 (first 504 terms from V. Raman)
Index entries for linear recurrences with constant coefficients, order 251.

Cf. A000855, A126605, A000689.

[Modexp(2, n, 10000): n in [0..110]]; // Vincenzo Librandi, Aug 16 2016

PowerMod[2, Range[0, 100], 10000] (* Vincenzo Librandi, Aug 16 2016 *)

PARI
```
for(i=0,500,print(2^i%10000" "))
```

For n > 255: a(n) = a(n-1) - a(n-250) + a(n-251). - Ray Chandler, Aug 09 2025

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

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A000855 Final two digits of 2^n.

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A181610 The last n digits of powers of 2 start cycling. a(n) is the number of zero-free terms in this cycle.

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A216095 a(n) = 2^n mod 10000.

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