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 A309037 Exponential Demlo sequence, like 12345...54321, but for powers of 2 instead. 1
 2, 242, 24842, 2496842, 249936842, 24998736842, 2499974736842, 249999494736842, 24999989894736842, 2499999797894736842, 249999995957894736842, 24999999919157894736842, 2499999998383157894736842 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Lim_{n->infinity} a(n)/10^(2n-1) = 0.25, thus the first digits converge toward 24999999999999999999999... 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. For example, 24998736842 / 2 = 12499368421. 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. If 8421... is analytically continued, 052631578947436... is obtained because Sum_{i>=1} 1/(2^n*10^n) is 1/19. 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. LINKS Seiichi Manyama, Table of n, a(n) for n = 1..500 FORMULA 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). Conjectures from Colin Barker, Jul 16 2019: (Start) G.f.: 2*x*(1 - 10*x)*(1 + 10*x) / ((1 - x)*(1 - 20*x)*(1 - 100*x)). a(n) = (-80 - 3*4^n*5^(1+n) + 19*100^n) / 760. a(n) = 121*a(n-1) - 2120*a(n-2) + 2000*a(n-3) for n>3. (End) EXAMPLE For n = 4:   2000000    8 - 2 = 6    400000     80000     16000    4 - 1 = 3       800        40   +     2   -------   2496842 For n = 12: 2*10^(24-2) + 4*10^(24-3) + 8*10^(24-4) + ... + 4096*10^11 + ... + 8*10^2 + 4*10^1 + 2   20000000000000000000000    24 - 2 = 22    4000000000000000000000     800000000000000000000     160000000000000000000      32000000000000000000       6400000000000000000       1280000000000000000        256000000000000000         51200000000000000         10240000000000000          2048000000000000           409600000000000    12 - 1 = 11            20480000000000             1024000000000               51200000000                2560000000                 128000000                   6400000                    320000                     16000                       800                        40   +                     2   -----------------------   24999999919157894736842 CROSSREFS Cf. A002477, A000079. Numbers produced from A000079 using A002477 algorithm. Sequence in context: A074256 A146312 A109930 * A013509 A013472 A013505 Adjacent sequences:  A309034 A309035 A309036 * A309038 A309039 A309040 KEYWORD base,nonn AUTHOR Eliora Ben-Gurion, Jul 08 2019 STATUS approved

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Last modified February 26 19:41 EST 2020. Contains 332295 sequences. (Running on oeis4.)