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%I #16 Aug 01 2021 17:43:07
%S 3,12,53,254,1255,6256,31257,156258,781259,3906260,19531261,97656262,
%T 488281263,2441406264,12207031265,61035156266,305175781267,
%U 1525878906268,7629394531269,38146972656270,190734863281271
%N Consider the final n decimal digits of 2^j for all values of j. They are periodic. Sequence gives position (or phase) of the maximal value seen in these n digits.
%C The last n digits of 2^a(n) are predictable if maximal values of periods are known.
%H Harvey P. Dale, <a href="/A060460/b060460.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Fi#final">Index entries for sequences related to final digits of numbers</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7, -11, 5).
%F a(1) = 3, a(n) = 5*a(n-1)-(3+4*(n-2)).
%F a(n) = a(n) = 2*5^(n-1) + n.
%F G.f.: (-3 + 9 x - 2 x^2)/((-1 + x)^2 (-1 + 5 x)) - _Harvey P. Dale_, Aug 01 2021
%e a(2) = 5*3-(3+4*0) = 15-3 = 12, etc...
%e For n=2, the last 2 digits of powers of 2 have the period {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} displayed in A000855. The maximum is 96 and it occurs at 2^12=4096. So a(2)=12.
%t nxt[{n_,a_,b_}]:={n+1,b,5b-(3+4(n-1))}; NestList[nxt,{2,3,12},20][[All,2]] (* or *) Table[2*5^(n-1)+n,{n,30}] (* or *) LinearRecurrence[{7,-11,5},{3,12,53},30] (* _Harvey P. Dale_, Aug 01 2021 *)
%Y Cf. A000079, A000855, A005054, A060458.
%K base,nonn
%O 1,1
%A _Labos Elemer_, Apr 09 2001
%E Offset 1 (and formulas adapted) from _Michel Marcus_, Mar 25 2020
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