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A378360 a(1) = 0, a(n) = Sum_{digits d in a(n-1)} (d+[d==0])*c(d,n-1) where c(d,k) = number of digits d in a(1..a(k)), with Iverson bracket.

%I A378360 #12 Nov 27 2024 18:36:07
%S A378360 0,1,1,2,2,4,4,8,8,16,9,9,18,28,38,43,18,53,14,22,10,9,27,19,44,24,42,
%T A378360 48,92,63,24,60,21,31,25,34,62,50,19,65,50,30,27,42,78,85,102,51,48,
%U A378360 132,72,64,92,101,24,100,27,77,49,136,87,144,91,101,33,33,39
%N A378360 a(1) = 0, a(n) = Sum_{digits d in a(n-1)} (d+[d==0])*c(d,n-1) where c(d,k) = number of digits d in a(1..a(k)), with Iverson bracket.
%C A378360 This sequence is a version of the "paint sprayer" sequence A279818 that engages all decimal digits, including d = 0. For each digit d in a(n-1), a(n) records the sum of all same digits d observed in terms a(i); i = 1..n-1, summed over every d in a(n-1), wherein each occurrence of digit d = 0 in a(n-1) and prior terms is counted as 1 instead of 0.
%C A378360 The sums form 2^10 or 1024 trajectories similar to those in A279818, which exhibits 2^9 trajectories, since digit 0 has no effect on the sums in that sequence.
%C A378360 Let signature S be the set of digits in the number m; for reference we may order the digits least to greatest. For example, for m = 90210, S(90210) = {0, 1, 2, 9}. Signature S may be compactified via the sum of 2^k for k in S. Thus, for example, S(90210) compactified is 2^0 + 2^1 + 2^2 + 2^9 = 1+2+4+512 = 521.
%C A378360 Let trajectory T(S) be that associated with S. In scatterplot, the trajectories are arranged such that the smaller compactified signatures have the least "slope" while the pandigital signature 1023 has the greatest slope.
%C A378360 This trajectory arrangement is present and clearer in A279818, but those signatures involving 0 (odd compactified signatures) in this sequence are skewed, since 0 cannot begin a decimal number m > 0, and thereby has a smaller frequency.
%C A378360 Compared to A279818, the scatterplot appears to have denser trajectories.
%C A378360 The number 5 cannot appear in the sequence for n > 18.
%C A378360 First pandigital number is a(28035151) = 1053287964.
%C A378360 The signature S = {5} has not appeared for n < 10^8. It will first appear for some repdigit where the repeated digit is d = 5.
%H A378360 Michael De Vlieger, <a href="/A378360/b378360.txt">Table of n, a(n) for n = 1..10000</a>
%H A378360 Michael De Vlieger, <a href="/A378360/a378360.png">Scatterplot of a(n)</a>, n = 1..10^6.
%H A378360 Michael De Vlieger, <a href="/A378360/a378360_1.png">Log log scatterplot of c(d, n)</a>, n = 1..10^5, d = 0..9, with a color function showing d = 0 in black, d = 1 in red, d = 2 in orange, ..., d = 9 in purple.
%H A378360 Michael De Vlieger, <a href="/A378360/a378360_1.txt">First emergence of digit signatures in A378360</a> and for versions of this sequence in bases b = 2..9. Includes remarks for each base, proof that a(n) = 5 is impossible for n > 18 in decimal, and related code.
%e A378360 Let c(d) be the number of digits d in the sequence a(1..n-1), abbreviating the dyadic function for concision:
%e A378360 a(2) = 1 since 0+[0==0]*c(0) = 1*1 = 1.
%e A378360 a(3) = 1 since 1+[1==0]*c(1) = 1*1 = 1.
%e A378360 a(4) = 2 since 1*c(1) = 1*2 = 2.
%e A378360 a(5) = 2 since 2*c(2) = 2*1 = 2.
%e A378360 a(6) = 4 since 2*c(2) = 2*2 = 4.
%e A378360 ...
%e A378360 a(10) = 16 since a(9) = 8, and 8*c(8) = 8*2 = 16.
%e A378360 a(11) = 9 since 1*c(1)+6*c(6) = 1*3+6*1 = 9.
%e A378360 ...
%e A378360 a(19) = 14 since a(18) = 53, and 5*c(5)+3*c(3) = 5*1+3*3 = 14.
%e A378360 a(20) = 22 since 1*c(1)+4*c(4) = 1*6+4*4 = 22.
%e A378360 a(21) = 10 since 2*c(2) = 2*5 = 10, (not 2*c(2)+2*c(2) = 20), etc.
%t A378360 nn = 120; a[1] = j = 0; c[_] := 0;
%t A378360 Do[k = Total@ Map[(#1 + Boole[#1 == 0])*(c[#1] += #2) & @@ # &,
%t A378360     Tally@ IntegerDigits[j] ];
%t A378360   Set[{a[n], j}, {k, k}],
%t A378360 {n, 2, nn}]; Array[a, nn]
%Y A378360 Cf. A279818, A378316, A378359.
%K A378360 nonn,easy
%O A378360 1,4
%A A378360 _Michael De Vlieger_ and _David James Sycamore_, Nov 26 2024