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A309037 Exponential Demlo sequence, like 12345...54321, but for powers of 2 instead.

%I A309037 #39 Jul 04 2023 14:01:55
%S A309037 2,242,24842,2496842,249936842,24998736842,2499974736842,
%T A309037 249999494736842,24999989894736842,2499999797894736842,
%U A309037 249999995957894736842,24999999919157894736842,2499999998383157894736842
%N A309037 Exponential Demlo sequence, like 12345...54321, but for powers of 2 instead.
%C A309037 Lim_{n->infinity} a(n)/10^(2n-1) = 0.25, thus the first digits converge toward 24999999999999999999999...
%C A309037 In other words, Sum_{i>=1} 2^n/10^n = Sum_{i>=1} 5^(-n) = 5/(1-5) = 5/4 = 1.25. Excluding the 1 at the beginning of the number gives 0.25. Dividing each term by 2 gives the previous term with 1s attached on each side.
%C A309037 For example, 24998736842 / 2 = 12499368421.
%C A309037 In the set of {a(n)}, the final digits of a(n) eventually tend to be the repeating portion of 1/19 as n approaches infinity: ... 052631578947368421 05263157894736842.
%C A309037 If 8421... is analytically continued, 052631578947436... is obtained because Sum_{i>=1} 1/(2^n*10^n) is 1/19.
%C A309037 I propose that the Demlo function should be generalized, so that the function A002477(A000079(n)) produces this sequence. As another example, A002477(A000040(n)) should produce 2, 232, 23532, 2357532, 235817532, 23582417532, etc.
%H A309037 Seiichi Manyama, <a href="/A309037/b309037.txt">Table of n, a(n) for n = 1..500</a>
%H A309037 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (121, -2120, 2000).
%F A309037 a(n) = 2^1*10^0 + 2^2*10^1 + ... + 2^(n-1)*10^(n-2) + 2^n*10^(n-1) + 2^(n-1)*10^n + 2^(n-2)*10^(n+1) + ... + 2^2*10^(2n-3) + 2^1*10^(2n-2).
%F A309037 Conjectures from _Colin Barker_, Jul 16 2019: (Start)
%F A309037 G.f.: 2*x*(1 - 10*x)*(1 + 10*x) / ((1 - x)*(1 - 20*x)*(1 - 100*x)).
%F A309037 a(n) = (-80 - 3*4^n*5^(1+n) + 19*100^n) / 760.
%F A309037 a(n) = 121*a(n-1) - 2120*a(n-2) + 2000*a(n-3) for n>3.
%F A309037 (End)
%e A309037 For n = 4:
%e A309037   2000000    8 - 2 = 6
%e A309037    400000
%e A309037     80000
%e A309037     16000    4 - 1 = 3
%e A309037       800
%e A309037        40
%e A309037   +     2
%e A309037   -------
%e A309037   2496842
%e A309037 For n = 12:
%e A309037 2*10^(24-2) + 4*10^(24-3) + 8*10^(24-4) + ... + 4096*10^11 + ... + 8*10^2 + 4*10^1 + 2
%e A309037   20000000000000000000000    24 - 2 = 22
%e A309037    4000000000000000000000
%e A309037     800000000000000000000
%e A309037     160000000000000000000
%e A309037      32000000000000000000
%e A309037       6400000000000000000
%e A309037       1280000000000000000
%e A309037        256000000000000000
%e A309037         51200000000000000
%e A309037         10240000000000000
%e A309037          2048000000000000
%e A309037           409600000000000    12 - 1 = 11
%e A309037            20480000000000
%e A309037             1024000000000
%e A309037               51200000000
%e A309037                2560000000
%e A309037                 128000000
%e A309037                   6400000
%e A309037                    320000
%e A309037                     16000
%e A309037                       800
%e A309037                        40
%e A309037   +                     2
%e A309037   -----------------------
%e A309037   24999999919157894736842
%Y A309037 Cf. A002477, A000079. Numbers produced from A000079 using A002477 algorithm.
%K A309037 base,nonn
%O A309037 1,1
%A A309037 _Eliora Ben-Gurion_, Jul 08 2019