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%I #38 Sep 08 2022 08:46:06
%S 0,1,3,6,7,9,12,16,18,21,25,30,33,37,42,48,49,51,54,58,60,63,67,72,75,
%T 79,84,90,94,99,105,112,114,117,121,126,129,133,138,144,148,153,159,
%U 166,171,177,184,192,195,199,204,210,214,219,225,232,237,243,250,258,264,271,279,288,289,291,294,298,300,303,307,312,315,319,324,330,334,339,345,352,354,357
%N a(n) = Sum_{i=0..n} digsum_4(i), where digsum_4(i) = A053737(i).
%D Jean-Paul Allouche and Jeffrey Shallit, Automatic sequences, Cambridge University Press, 2003, p. 94.
%H Reinhard Zumkeller, <a href="/A231664/b231664.txt">Table of n, a(n) for n = 0..10000</a>
%H Jean Coquet, <a href="https://doi.org/10.1016/0022-314X(86)90067-3">Power sums of digital sums</a>, J. Number Theory, Vol. 22, No. 2 (1986), pp. 161-176.
%H P. J. Grabner, P. Kirschenhofer, H. Prodinger and R. F. Tichy, <a href="http://math.sun.ac.za/~hproding/abstract/abs_80.htm">On the moments of the sum-of-digits function</a>, <a href="http://math.sun.ac.za/~hproding/pdffiles/st_andrews.pdf">PDF</a>, Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), pp. 263-271, Kluwer Acad. Publ., Dordrecht, 1993.
%H Hsien-Kuei Hwang, Svante Janson and Tsung-Hsi 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, Vol. 13, No. 4 (2017), Article #47; <a href="https://www.researchgate.net/profile/Hsien-Kuei-Hwang/publication/320642171_Exact_and_Asymptotic_Solutions_of_a_Divide-and-Conquer_Recurrence_Dividing_at_Half_Theory_and_Applications/links/59f9a5be0f7e9b553ec0eaad">ResearchGate link</a>; <a href="http://140.109.74.92/hk/wp-content/files/2016/12/aat-hhrr-1.pdf">preprint</a>, 2016.
%H J.-L. Mauclaire and Leo Murata, <a href="https://dx.doi.org/10.3792/pjaa.59.274">On q-additive functions. I</a>, Proc. Japan Acad. Ser. A Math. Sci., Vol. 59, No. 6 (1983), pp. 274-276.
%H J.-L. Mauclaire and Leo Murata, <a href="https://dx.doi.org/10.3792/pjaa.59.441">On q-additive functions. II</a>, Proc. Japan Acad. Ser. A Math. Sci., Vol. 59, No. 9 (1983), pp. 441-444.
%H J. R. Trollope, <a href="http://www.jstor.org/stable/2687954">An explicit expression for binary digital sums</a>, Math. Mag., Vol. 41, No. 1 (1968), pp. 21-25.
%F G.f.: g(x) satisfies g(x) = (1+x+x^2+x^3)^2*g(x^4) + (x+2*x^2+3*x^3)/(1-x-x^4+x^5). - _Robert Israel_, Sep 20 2017
%F a(n) ~ 3*n*log(n)/(4*log(2)). - _Amiram Eldar_, Dec 09 2021
%p ListTools:-PartialSums([seq(convert(convert(n,base,4),`+`), n=0..200)]); # _Robert Israel_, Sep 20 2017
%t Table[Sum[Total[IntegerDigits[j, 4]], {j,0,n}], {n, 0, 100}] (* _G. C. Greubel_, Feb 16 2019 *)
%o (PARI) a(n) = sum(i=0, n, sumdigits(i, 4)); \\ _Michel Marcus_, Sep 20 2017
%o (Magma) [(&+[&+Intseq(j, 4): j in [0..n]]): n in [0..100]]; // _G. C. Greubel_, Feb 16 2019
%Y Cf. A053737, A231665, A231666, A231667.
%K nonn,base
%O 0,3
%A _N. J. A. Sloane_, Nov 13 2013
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