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%I #40 Sep 08 2022 08:45:07
%S 0,1,5,14,30,55,91,140,204,285,286,290,299,315,340,376,425,489,570,
%T 670,674,683,699,724,760,809,873,954,1054,1175,1184,1200,1225,1261,
%U 1310,1374,1455,1555,1676,1820,1836,1861,1897,1946,2010,2091,2191,2312,2456
%N a(n) = a(n-1) + square of the sum of digits of n.
%C a(n) = Sum_{i=0..n} digsum(i)^2, where digsum(i) = A007953(i). - _N. J. A. Sloane_, Nov 13 2013
%H Amiram Eldar, <a href="/A074784/b074784.txt">Table of n, a(n) for n = 0..10000</a> (terms 1..990 from Indranil Ghosh)
%H Tom C. Brown, <a href="https://www.fq.math.ca/Scanned/32-3/brown.pdf">Powers of Digital Sums</a>, The Fibonacci Quarterly, Vol. 32, No. 3 (1994), pp. 207-210.
%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), Kluwer Acad. Publ., Dordrecht, 1993, pp. 263-271; <a href="https://math.sun.ac.za/prodinger/pdffiles/st_andrews.pdf">alternative link</a>.
%H Robert E. Kennedy and Curtis N. Cooper, <a href="http://www.fq.math.ca/Scanned/29-2/kennedy.pdf">An extension of a theorem by Cheo and Yien concerning digital sums</a>, Fibonacci Quarterly, Vol. 29, No. 2 (1991), pp. 145-149.
%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 Harald Riede, <a href="http://www.fq.math.ca/Scanned/36-1/riede.pdf">Asymptotic estimation of a sum of digits</a>, Fibonacci Quarterly, Vol. 36, No. 1 (1998), pp. 72-75.
%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 a(n) = Sum_{k=1..n} s(k)^2 = Sum_{k=1..n} A007953(k)^2, where s(k) denotes the sum of the digits of k in decimal representation.
%F Asymptotic expression: a(n-1) = Sum_{k=1..n-1} s(k)^2 = 20.25*n*log_10(n)^2 + O(n*log_10(n)).
%F In general: Sum_{k=1..n-1} s(k)^m = n*((9/2)*log_10(n))^m + O(n*log_10(n)^(m-1)).
%p See A037123.
%t Accumulate @ Array[(Plus @@ IntegerDigits[#])^2 &, 50] (* _Amiram Eldar_, Jan 20 2022 *)
%o (Magma) [n eq 1 select n else Self(n-1)+(&+Intseq(n))^2: n in [1..48]]; // _Bruno Berselli_, Jul 12 2011
%Y Cf. A007953, A037123, A231688, A231689, A254524.
%Y Partial sums of A118881.
%K nonn,base
%O 0,3
%A Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 07 2002
%E Offset changed to 0 and a(0) prepended by _Amiram Eldar_, Jan 20 2022
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