A060458 Maximum value seen in the final n decimal digits of 2^j for all values of j.

8, 96, 992, 9984, 99968, 999936, 9999872, 99999744, 999999488, 9999998976, 99999997952, 999999995904, 9999999991808, 99999999983616, 999999999967232, 9999999999934464, 99999999999868928, 999999999999737856, 9999999999999475712, 99999999999998951424, 999999999999997902848

Offset

1,1

Comments

Consider the final n decimal digits of 2^j for all values of j. They are periodic. Sequence gives maximal value seen in these n digits.

With f(n) = a(n+1) - a(n), the difference f(n) - a(n) is always 8*10^n meaning that a(n) becomes its own "first differences" sequence when each term is prefixed a digit '8'. For higher order differences, the prefix 8 becomes: 8*10^n*Sum_{k=0..m-1} 9^k where m is the order. - R. J. Cano, May 11 2014

Links

Table of n, a(n) for n=1..21.

Index entries for sequences related to final digits of numbers.

Index entries for linear recurrences with constant coefficients, signature (12,-20).

Formula

a(n) = 10^n - 2^n = 2^n*(5^n - 1).

From Geoffrey Critzer, Dec 15 2011: (Start)

a(n) = 12*a(n-1) - 20*a(n-2).

O.g.f.: 1/(1-10*x) - 1/(1-2*x). (End)

a(n) = f(n,0) where f(x,y) = Sum_{j=0..x+y-1} (2^(3*x-2*j)*binomial(x,j)). - R. J. Cano, May 15 2014

a(n) = 2^(n+2)*A003463(n). - R. J. Cano, Sep 25 2014

a(n) = 8*A016134(n-1). - R. J. Mathar, Mar 10 2022

E.g.f.: exp(2*x)*(exp(8*x) - 1). - Elmo R. Oliveira, Mar 26 2025

Example

Maximum of the last 4 digits of powers of 2 is 9984=10000-16. It occurs at 2^254. 2^254 = 289480223.....01978282409984 (with 77 digits, last 4 ones are ...9984). The period length of the last-4-digit segment is A005054(4)=500. For n=4 period: amplitude=9984, phase=254.

Maple

A060458:=n->10^n-2^n: seq(A060458(n), n=1..20); # Wesley Ivan Hurt, Sep 25 2014

Mathematica

RecurrenceTable[{a[n] == 12 a[n - 1] - 20 a[n - 2], a[0] == 0, a[1] == 8}, a[n], {n, 1, 20}]  (* Geoffrey Critzer, Dec 15 2011*)

Prog

(SageMath) [10^n - 2^n for n in range(1, 19)] # Zerinvary Lajos, Jun 05 2009
(PARI) a(n)=sum(j=0, n-1, 2^(3*n-2*j)*binomial(n, j)) \\ R. J. Cano, May 15 2014
(Magma) [10^n-2^n : n in [1..20]]; // Wesley Ivan Hurt, Sep 25 2014
(PARI) A060458(n)=(5^n-1)<<n \\ M. F. Hasler, Oct 31 2014

Crossrefs

Cf. A000079, A003463, A005054, A060460, A016134.

Sequence in context: A276593 A099675 A211345 * A173834 A260627 A098430

Adjacent sequences: A060455 A060456 A060457 * A060459 A060460 A060461

Keyword

base,nonn,easy

Author

Labos Elemer, Apr 09 2001

Extensions

Edited by M. F. Hasler, Oct 31 2014

More terms from Elmo R. Oliveira, Mar 26 2025

Status

approved