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A074784
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a(n) = a(n-1) + square of the sum of digits of n.
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5
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0, 1, 5, 14, 30, 55, 91, 140, 204, 285, 286, 290, 299, 315, 340, 376, 425, 489, 570, 670, 674, 683, 699, 724, 760, 809, 873, 954, 1054, 1175, 1184, 1200, 1225, 1261, 1310, 1374, 1455, 1555, 1676, 1820, 1836, 1861, 1897, 1946, 2010, 2091, 2191, 2312, 2456
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OFFSET
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0,3
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COMMENTS
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LINKS
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P. J. Grabner, P. Kirschenhofer, H. Prodinger and R. F. Tichy, On the moments of the sum-of-digits function, PDF, Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), Kluwer Acad. Publ., Dordrecht, 1993, pp. 263-271; alternative link.
J.-L. Mauclaire and Leo Murata, On q-additive functions. I, Proc. Japan Acad. Ser. A Math. Sci., Vol. 59, No. 6 (1983), pp. 274-276.
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FORMULA
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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.
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)).
In general: Sum_{k=1..n-1} s(k)^m = n*((9/2)*log_10(n))^m + O(n*log_10(n)^(m-1)).
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MAPLE
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MATHEMATICA
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Accumulate @ Array[(Plus @@ IntegerDigits[#])^2 &, 50] (* Amiram Eldar, Jan 20 2022 *)
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PROG
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(Magma) [n eq 1 select n else Self(n-1)+(&+Intseq(n))^2: n in [1..48]]; // Bruno Berselli, Jul 12 2011
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 07 2002
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EXTENSIONS
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Offset changed to 0 and a(0) prepended by Amiram Eldar, Jan 20 2022
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STATUS
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approved
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