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%I #72 Jan 28 2023 18:24:20
%S 1210,2020,21200,3211000,42101000,521001000,6210001000,52200100019,
%T 52200100108,52200101007,52200110006,52201100004,52210100003,
%U 53010100019,53010100108,53010101007,53010110006,53011100004,53110100002,61200020006,62200010001,63010010001,70200002007,72100001000,431110000299
%N Autobiographical numbers: the first digit of the term counts how many 0's the term in total has, the second how many 1's etc. up to the last digit but no more than b-1, where b is the base of the sequence. This sequence is in base b=10.
%C Other terms include: 640010100011, 722000010001, 722000010010, 730100010001, 730100010010, 802000002008, 802000002080, 821000001000, 6500011000111, 7400100100011, 7400100100101, 8301000010001, 9020000002009, 9020000002090, 9020000002900, 9210000001000.
%C This sequence is finite. The last term starts with 99999999898... and has 89 digits.
%C This sequence is for base b=10. For each base b > 2, the last term of the corresponding sequence has b^2 - b - 1 digits.
%C For b > 2, the final term of the sequence equals b^((b-1)^2 - 2) - (b-1)*b^((b-1)*b - 1) + ((b^(b-1) - 1)/(b-1)) * b^(b-1) * ((b-1)*b^(b*(b-1)) - b^((b-1)^2 + 1) + 1)/(b^(b-1) - 1)^2. The base-b expansion of this number is the concatenation of b-2 digits b-1, 1 digit b-2, 1 digit b-1, b-3 digits b-2, and b-1 digits k for each k in b-3..0. This is equivalent to taking a string of digits consisting of b-1 copies of every valid base-b digit (0..b-1), sorting its digits in descending order, removing one of the digits b-2, and then swapping the positions of the last digit b-1 and the first digit b-2. (Thus, for b=10, the base-10 expansion of the final term is the concatenation of eight 9's, one 8, one 9, seven 8's, nine 7's, nine 6's, ..., nine 1's, and nine 0's.) - _Jon E. Schoenfield_, Nov 21 2022
%H Michael S. Branicky, <a href="/A358538/b358538.txt">Table of n, a(n) for n = 1..10000</a>
%H Michael S. Branicky, <a href="/A358538/a358538.py.txt">Python program</a>
%e 63010010001 is a term: we have six 0's, three 1's, one 3 and one 6 as digits in the term, visualized as follows:
%e Digits: 0123456789
%e term: 63010010001.
%e Note that this example also shows, starting from the 11th digit, there is no more representation of the frequency of that digit, because only the first b digits of its base-b expansion count the occurrences of the corresponding digit. In this case, the last digit, 1, is the 11th.
%o (Python) # see linked program
%Y Cf. A046043, A138480.
%K nonn,fini,base
%O 1,1
%A _Marc Morgenegg_, Nov 21 2022
%E a(14)-a(19) inserted and a(25) from _Michael S. Branicky_, Nov 21 2022
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