|
%I #39 Jul 04 2023 14:01:55
%S 2,242,24842,2496842,249936842,24998736842,2499974736842,
%T 249999494736842,24999989894736842,2499999797894736842,
%U 249999995957894736842,24999999919157894736842,2499999998383157894736842
%N Exponential Demlo sequence, like 12345...54321, but for powers of 2 instead.
%C Lim_{n->infinity} a(n)/10^(2n-1) = 0.25, thus the first digits converge toward 24999999999999999999999...
%C 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 For example, 24998736842 / 2 = 12499368421.
%C 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 If 8421... is analytically continued, 052631578947436... is obtained because Sum_{i>=1} 1/(2^n*10^n) is 1/19.
%C 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 Seiichi Manyama, <a href="/A309037/b309037.txt">Table of n, a(n) for n = 1..500</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (121, -2120, 2000).
%F 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 Conjectures from _Colin Barker_, Jul 16 2019: (Start)
%F G.f.: 2*x*(1 - 10*x)*(1 + 10*x) / ((1 - x)*(1 - 20*x)*(1 - 100*x)).
%F a(n) = (-80 - 3*4^n*5^(1+n) + 19*100^n) / 760.
%F a(n) = 121*a(n-1) - 2120*a(n-2) + 2000*a(n-3) for n>3.
%F (End)
%e For n = 4:
%e 2000000 8 - 2 = 6
%e 400000
%e 80000
%e 16000 4 - 1 = 3
%e 800
%e 40
%e + 2
%e -------
%e 2496842
%e For n = 12:
%e 2*10^(24-2) + 4*10^(24-3) + 8*10^(24-4) + ... + 4096*10^11 + ... + 8*10^2 + 4*10^1 + 2
%e 20000000000000000000000 24 - 2 = 22
%e 4000000000000000000000
%e 800000000000000000000
%e 160000000000000000000
%e 32000000000000000000
%e 6400000000000000000
%e 1280000000000000000
%e 256000000000000000
%e 51200000000000000
%e 10240000000000000
%e 2048000000000000
%e 409600000000000 12 - 1 = 11
%e 20480000000000
%e 1024000000000
%e 51200000000
%e 2560000000
%e 128000000
%e 6400000
%e 320000
%e 16000
%e 800
%e 40
%e + 2
%e -----------------------
%e 24999999919157894736842
%Y Cf. A002477, A000079. Numbers produced from A000079 using A002477 algorithm.
%K base,nonn
%O 1,1
%A _Eliora Ben-Gurion_, Jul 08 2019
| 