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%I #23 Sep 08 2022 08:45:06
%S 0,1,3,6,7,8,9,10,12,14,16,18,21,24,27,30,31,32,33,34,35,36,37,38,39,
%T 40,41,42,43,44,45,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,
%U 81,84,87,90,93,96,99,102,105,108,111,114,117,120,123,126,127,128,129,130
%N Cumulative sum of initial digits of (n base 4).
%H Hsien-Kuei Hwang, S. Janson, T.-H. Tsai, <a href="http://140.109.74.92/hk/wp-content/files/2016/12/aat-hhrr-1.pdf">Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications</a>, Preprint 2016.
%H Hsien-Kuei Hwang, S. Janson, T.-H. Tsai, <a href="https://doi.org/10.1145/3127585">Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications</a>, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585.
%F a(n) = Sum_{i=0..n} A000030(A007090(n)).
%F a(n) = Sum_{i=0..n} first-digit(i base 4) where (i base 4) = A007090(i).
%e n in init cumulative
%e n base 4 dgt sum
%e - ------ ---- ----------
%e 0 0 0 0
%e 1 1 1 1
%e 2 2 2 3
%e 3 3 3 6
%e 4 10 1 7
%e 5 11 1 8
%o (Magma) [0] cat [&+[Reverse(Intseq(k,4))[1]:k in [1..n]]:n in [1..70]]; // _Marius A. Burtea_, Dec 19 2019
%Y Cf. A000030, A007090, A109453.
%K base,easy,nonn
%O 0,3
%A _Jonathan Vos Post_, Aug 28 2005