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A074784 a(n) = a(n-1) + square of the sum of digits of n. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n) = Sum_{i=0..n} digsum(i)^2, where digsum(i) = A007953(i). - N. J. A. Sloane, Nov 13 2013

REFERENCES

Grabner, P. J.; Kirschenhofer, P.; Prodinger, H.; Tichy, R. F.; On the moments of the sum-of-digits function. Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), 263-271, Kluwer Acad. Publ., Dordrecht, 1993.

LINKS

Indranil Ghosh, Table of n, a(n) for n = 0..990

J. Coquet, Power sums of digital sums, J. Number Theory 22 (1986), no. 2, 161-176.

R. E. Kennedy and C. N. Cooper, An extension of a theorem by Cheo and Yien concerning digital sums, Fibonacci Q. 29, No. 2, 145-149 (1991).

J.-L. Mauclaire and Leo Murata, On q-additive functions, I. Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 6, 274-276.

J.-L. Mauclaire and Leo Murata, On q-additive functions, II. Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 9, 441-444.

H. Riede, Asymptotic estimation of a sum of digits, Fibonacci Q. 36, No. 1, 72-75 (1998).

J. R. Trollope, An explicit expression for binary digital sums, Math. Mag. 41 1968 21-25.

FORMULA

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)*log10(n))^m + O(n*log_10(n)^(m-1)).

MAPLE

See A037123.

PROG

(MAGMA) [n eq 1 select n else Self(n-1)+(&+Intseq(n))^2: n in [1..48]];  // Bruno Berselli, Jul 12 2011

CROSSREFS

Cf. A007953, A037123, A231688, A231689, A254524.

Partial sums of A118881.

Sequence in context: A231677 A231681 A231685 * A109678 A000330 A266783

Adjacent sequences:  A074781 A074782 A074783 * A074785 A074786 A074787

KEYWORD

nonn,base

AUTHOR

Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 07 2002

STATUS

approved

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Last modified May 30 03:33 EDT 2017. Contains 287305 sequences.