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Single-digit numbers in the order in which they first appear in the decimal expansions of powers of 2, followed by the two-digit numbers in the order in which they appear, then the three-digit numbers, and so on.
+10
4
1, 2, 4, 8, 6, 3, 5, 0, 9, 7, 16, 32, 64, 12, 28, 25, 56, 51, 10, 24, 20, 48, 40, 96, 81, 19, 92, 63, 38, 84, 27, 76, 68, 65, 55, 53, 36, 13, 31, 72, 26, 62, 21, 14, 44, 52, 42, 88, 85, 57, 97, 71, 15, 41, 94, 43, 30, 83, 86, 60, 67, 77, 33, 35, 54, 34, 17, 45
COMMENTS
Apparently this algorithm applied to most sequences will produce a fractal scatterplot graph. - David Williams, Jan 20 2019
EXAMPLE
1,2,4,8,16,32,64,128,256,512,1024, ..., 4096, ..., 32768, ... gives 1,2,4,8,6,3,5,0,9,7.
Then we get 16,32,64,12,28,25,56,51,10,24,20,48,40,96,81,19,92,...
11 does not appear until 2^40 = 1099511627776.
PROG
(PARI) See Links section.
Single-digit numbers in the order in which they first appear in the decimal expansion of e, followed by the two-digit numbers in the order in which they appear, then the three-digit numbers, and so on.
+10
2
2, 7, 1, 8, 4, 5, 9, 0, 3, 6, 27, 71, 18, 82, 28, 81, 84, 45, 59, 90, 52, 23, 35, 53, 36, 60, 87, 74, 47, 13, 26, 66, 62, 24, 49, 97, 77, 75, 57, 72, 70, 93, 69, 99, 95, 96, 67, 76, 40, 63, 30, 54, 94, 38, 21, 17, 78, 85, 25, 51, 16, 64, 42, 46, 39, 91, 19
COMMENTS
Note that (except for 0 itself), numbers may not begin with 0. So that when we reach ...459045..., this contributes 90 to the sequence but not "04". - N. J. A. Sloane, Feb 08 2017
EXAMPLE
Consider the decimal expansion of e=2.718281828459045235360...
The first 4 terms are 2,7,1,8 since these single digits appear in that order above. We do not encounter a different digit till we reach 4,5,9,0, thus these follow the first four in the sequence. We encounter 3 next, and finally 6 and have found all the single digits in the expansion.
a(11)=27 because we find the two-digit group "27" first, followed by a(12)=71, etc. until we exhaust the 90 possible two-digit groups that do not start with a zero.
a(101)=271 because we find the three-digit group "271" first, followed by a(102)=718, etc. until we exhaust the 900 possible 3-digit groups that do not have leading zeros, etc. (End)
MATHEMATICA
e = First@ RealDigits@ N[E, 10^6]; MapIndexed[10^(First@ #2 - 1) - 1 - Boole[First@ #2 == 1] + Flatten@ Values@ KeySort@ PositionIndex@ #1 &, Table[SequencePosition[e, IntegerDigits@ k][[1, 1]], {n, 4}, {k, If[n == 1, 0, 10^(n - 1)], 10^n - 1}]] (* Michael De Vlieger, Feb 09 2017, Version 10.1 *)
EXTENSIONS
a(5), a(6), a(9), and a(10) inserted by Bobby Jacobs, Feb 09 2017
Digits in the order in which they appear in the fractional part of the decimal expansion of Pi.
+10
0
1, 4, 5, 9, 2, 6, 3, 8, 7, 0
COMMENTS
3,1,8,0,9,6,7,5,2,4 (see A049541) and 6,1,8,0,3,9,7,4,2,5 (see A094214) are the equivalent sequences for 1/Pi and 1/phi. Conjecture: These sequences are not random but are in ratio of 3/2 between the first six and last four digits and the first six digits and last four are the same.
MATHEMATICA
DeleteDuplicates[Rest[RealDigits[Pi, 10, 40][[1]]]] (* Harvey P. Dale, Jan 31 2020 *)
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