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A231665 a(n) = Sum_{i=0..n} digsum_4(i)^2, where digsum_4(i) = A053737(i).

%I A231665 #13 Jan 18 2019 16:12:58
%S A231665 0,1,5,14,15,19,28,44,48,57,73,98,107,123,148,184,185,189,198,214,218,
%T A231665 227,243,268,277,293,318,354,370,395,431,480,484,493,509,534,543,559,
%U A231665 584,620,636,661,697,746,771,807,856,920,929,945,970,1006,1022,1047,1083,1132,1157,1193,1242,1306,1342,1391,1455,1536,1537,1541,1550,1566,1570,1579,1595,1620,1629
%N A231665 a(n) = Sum_{i=0..n} digsum_4(i)^2, where digsum_4(i) = A053737(i).
%H A231665 Marius A. Burtea, <a href="/A231665/b231665.txt">Table of n, a(n) for n = 0..10000</a>
%H A231665 Jean Coquet, <a href="https://doi.org/10.1016/0022-314X(86)90067-3">Power sums of digital sums</a>, J. Number Theory 22 (1986), no. 2, 161-176.
%H A231665 P. J. Grabner, P. Kirschenhofer, H. Prodinger, 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), 263-271, Kluwer Acad. Publ., Dordrecht, 1993.
%H A231665 J.-L. Mauclaire, 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. 59 (1983), no. 6, 274-276.
%H A231665 J.-L. Mauclaire, 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. 59 (1983), no. 9, 441-444.
%H A231665 J. R. Trollope, <a href="http://www.jstor.org/stable/2687954">An explicit expression for binary digital sums</a>, Math. Mag. 41 1968 21-25.
%o A231665 (PARI) a(n) = sum(i=0, n, sumdigits(i, 4)^2); \\ _Michel Marcus_, Sep 20 2017
%o A231665 (MATLAB) for u=0:2000; v(u+1)=sum(dec2base(u,4)-'0');end
%o A231665 sol=cumsum(v.^2); % _Marius A. Burtea_, Jan 18 2019
%Y A231665 Cf. A053737, A231664, A231666, A231667.
%K A231665 nonn,base
%O A231665 0,3
%A A231665 _N. J. A. Sloane_, Nov 13 2013