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a(n) = Sum_{i=0..n} digsum_4(i)^3, where digsum_4(i) = A053737(i).
4

%I #13 Jan 18 2019 16:12:34

%S 0,1,9,36,37,45,72,136,144,171,235,360,387,451,576,792,793,801,828,

%T 892,900,927,991,1116,1143,1207,1332,1548,1612,1737,1953,2296,2304,

%U 2331,2395,2520,2547,2611,2736,2952,3016,3141,3357,3700,3825,4041,4384,4896,4923,4987,5112,5328,5392,5517,5733,6076,6201,6417,6760,7272,7488,7831,8343,9072,9073,9081,9108,9172

%N a(n) = Sum_{i=0..n} digsum_4(i)^3, where digsum_4(i) = A053737(i).

%H Marius A. Burtea, <a href="/A231666/b231666.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 22 (1986), no. 2, 161-176.

%H 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 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 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 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 (PARI) a(n) = sum(i=0, n, sumdigits(i, 4)^3); \\ _Michel Marcus_, Sep 20 2017

%o (MATLAB) for u=0:2000; v(u+1)=sum(dec2base(u,4)-'0'); end

%o sol=cumsum(v.^3); % _Marius A. Burtea_, Jan 18 2019

%Y Cf. A053737, A231664, A231665, A231667.

%K nonn,base

%O 0,3

%A _N. J. A. Sloane_, Nov 13 2013