%I #49 Sep 08 2022 08:45:03
%S 8,96,992,9984,99968,999936,9999872,99999744,999999488,9999998976,
%T 99999997952,999999995904,9999999991808,99999999983616,
%U 999999999967232,9999999999934464,99999999999868928,999999999999737856
%N Maximal value seen in the final n decimal digits of 2^j for all values of j.
%C 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.
%C 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
%H <a href="/index/Fi#final">Index entries for sequences related to final digits of numbers</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (12,-20).
%F a(n) = 10^n-2^n = 2^n*(5^n-1).
%F a(n) = 12*a(n-1) - 20*a(n-2); O.g.f.:1/(1-10x)-1/(1-2x). - _Geoffrey Critzer_, Dec 15 2011
%F a(n) = f(n,0); Given f(x,y)=sum{j=0..x+y-1}(2^(3*x-2*j)*binomial(x,j)). - _R. J. Cano_, May 15 2014
%F a(n) = 2^(n+2)*A003463(n). - _R. J. Cano_, Sep 25 2014
%F a(n) = 8*A016134(n-1). - _R. J. Mathar_, Mar 10 2022
%e 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.
%p A060458:=n->10^n-2^n: seq(A060458(n), n=1..20); # _Wesley Ivan Hurt_, Sep 25 2014
%t 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*)
%o (Sage) [10^n - 2^n for n in range(1,19)] # _Zerinvary Lajos_, Jun 05 2009
%o (PARI) a(n)=sum(j=0,n-1,2^(3*n-2*j)*binomial(n,j)) \\ _R. J. Cano_, May 15 2014
%o (Magma) [10^n-2^n : n in [1..20]]; // _Wesley Ivan Hurt_, Sep 25 2014
%o (PARI) A060458(n)=(5^n-1)<<n \\ _M. F. Hasler_, Oct 31 2014
%Y Cf. A000079, A003463, A005054, A060460, A016134.
%K base,nonn,easy
%O 1,1
%A _Labos Elemer_, Apr 09 2001
%E Edited by _M. F. Hasler_, Oct 31 2014