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A383967 Inventory sequence recording number of terms with 1,2,3,... decimal digits. Count until occurrence of a term = 0, whereupon reset the count; continue.

%I A383967 #21 Jun 07 2025 05:53:22
%S A383967 0,1,0,3,0,5,0,7,0,9,0,11,1,0,13,2,0,15,3,0,17,4,0,19,5,0,21,6,0,23,7,
%T A383967 0,25,8,0,27,9,0,29,10,0,30,12,0,31,14,0,32,16,0,33,18,0,34,20,0,35,
%U A383967 22,0,36,24,0,37,26,0,38,28,0,39,30,0,40,32,0,41,34
%N A383967 Inventory sequence recording number of terms with 1,2,3,... decimal digits. Count until occurrence of a term = 0, whereupon reset the count; continue.
%H A383967 Michael De Vlieger, <a href="/A383967/b383967.txt">Table of n, a(n) for n = 1..10000</a>
%H A383967 Michael De Vlieger, <a href="/A383967/a383967.png">Log log scatterplot of a(n)</a>, n = 1..2^20, showing a(n) = 0 instead as 1/2 for visibility.
%e A383967 a(1) = 0 because at first there are no terms with just one decimal digit. Following a zero term the count is reset and now since there is one term (a(1) = 0) with just one digit, a(2) = 1. Since there are no terms with two digits a(3) = 0. The count resets again and a(4) = 3 because there are now three terms (0,1,0) which have only one digit. Since there are no terms with two digits a(5) = 0.
%e A383967 The sequence continues 0,1,0,3,0,5,0,7,0,9,0 and at this point we have 11 terms with one digit then one term (11) with two digits, so the next two terms are 11,1 followed by 0 since there is not yet a term with three digits; and so on.
%t A383967 nn = 120; q[_] := 0; f[x_] := If[x == 0, 1, IntegerLength[x]]; j = 0; c = 1; q[1]++; {j}~Join~Reap[Do[If[j == 0, c = 1]; j = Sow[q[c]]; c++; q[f[j]]++, nn - 1] ][[-1, 1]] (* _Michael De Vlieger_, Jun 01 2025 *)
%Y A383967 Cf. A342585, A347738, A345730.
%K A383967 nonn,easy
%O A383967 1,4
%A A383967 _David James Sycamore_, May 16 2025