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%I #8 Feb 26 2019 08:17:56
%S 0,1,2,4,6,10,14,20,26,36,46,59,72,90,108,130,152,182,212,248,284,329,
%T 374,426,478,542,606,678,750,834,918,1011,1104,1214,1324,1446,1568,
%U 1708,1848,2002,2156,2333,2510,2702,2894,3108,3322,3552,3782,4040,4298,4575,4852,5156,5460,5784,6108,6464,6820,7196,7572,7977,8382
%N Indices where A028897(A322000(n)) increases. Partial sums of A072170(n,10).
%C A322000 lists all nonnegative integers m ordered by increasing "decibinary" value N = A028897(m) = Sum d[i]*2^i where d[i] are the decimal digits of m. A072170(N,10) says in how many ways a given N can be written in that way. Accordingly, this is also the length of runs of identical values A028897(A322000(k)), and the partial sums, listed here as a(k), give the indices of A322000 where the decibinary value of the terms go up by one.
%C We have a(k) <= A000123(k-1) with equality for 1 <= k <= 10: the first differences of A000123 give back that sequence with terms duplicated, and this is the limiting column of A072170.
%F a(n) = Sum_{0 <= k < n} A072170(k,10).
%o (PARI) A322003(n)=sum(k=0,n-1,A072170(k,10))
%o A322003_vec=vector(99,k,s=if(k>1,s)+A072170(k-1,10)) \\ more efficient for computing a large vector. Excludes the initial a(0) = 0 to have 1-based indices of the vector match the indices of the components a(n), n >= 1.
%Y Cf. A322000, A028897, A072170.
%K nonn
%O 0,3
%A _M. F. Hasler_, Feb 19 2019