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%I #28 Mar 12 2023 14:15:57
%S 0,1,21,321,4321,54321,654321,7654321,87654321,987654321,987654321,
%T 10987654321,210987654321,3210987654321,43210987654321,
%U 543210987654321,6543210987654321,76543210987654321,876543210987654321
%N Add (n mod 10)*10^(n-1) to the previous term, with a(0) = 0.
%C Original definition: "Concatenate next digit at left hand end."
%C This is misleading, since the concatenation of 0 yields the same term (leading zeros vanish), but upon the next concatenation of 1, the 0 reappears - except for a(1), which according to that description should equal a(1)=10: It is surprising that in this only case where the 0 is indeed present, it disappears upon left-concatenation of the digit 1! - _M. F. Hasler_, Jan 13 2013
%C From _Hieronymus Fischer_, Jan 23 2013: (Start)
%C A definition which is also consistent is: Start with terms 0 and 1 and then concatenate the next digit at the left hand end. If the next digit is a zero, keep this zero in mind so that the following digit is a 1 preceding a 0.
%C The sequence terms are the terms of A057137 in reversed digit order. Based on this understanding, the anomaly for the indices 0 and 1 where the terms are 0 and 1 instead of 0 and 10 (what one would expect) becomes self-explaining. Also, the special behavior when the zero digit is encountered becomes clear.
%C Examples: a(3) = 321 = Reversal(A057137(3)),
%C a(10) = 987654321 = Reversal(A057137(10)) = Reversal(1234567890). (End)
%H Hieronymus Fischer, <a href="/A057138/b057138.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = a(n-1) + 10^(n-1)*n - 10^n*floor(n/10) = A057139(n) mod 10^n.
%F a(n) = floor(((q/(10^10 - 1)) + q mod 10^(n mod 10))*10^(10*floor(n/10))), where q = 987654321. - _Hieronymus Fischer_, Jan 03 2013
%F G.f.: x(1-10(10x)^9 + 9(10x)^10)/((1-x) (1-10x)^2 (1-(10x)^10)). - _Robert Israel_, Jun 21 2017
%p ListTools:-PartialSums([seq((k mod 10)*10^(k-1), k=0..40)]); # _Robert Israel_, Jun 21 2017
%t Join[{c = 0}, Table[c = c + Mod[n, 10]*10^(n - 1), {n, 18}]] (* _T. D. Noe_, Jan 30 2013 *)
%o (PARI) a(n)=sum(i=0,n,i%10*10^(i-1)) \\ _M. F. Hasler_, Jan 13 2013
%Y Alternative progression for n >= 10 compared with A000422 and A014925.
%Y Cf. A057137 for reverse.
%K base,easy,nonn
%O 0,3
%A _Henry Bottomley_, Aug 12 2000
%E Better definition from _M. F. Hasler_, Jan 13 2013
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