A074784 a(n) = a(n-1) + square of the sum of digits of n.

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

Offset: 0

Author

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

Keywords

• nonn
• base

Comments

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

Links

• Amiram Eldar, Table of n, a(n) for n = 0..10000 (terms 1..990 from Indranil Ghosh)
• Tom C. Brown, Powers of Digital Sums, The Fibonacci Quarterly, Vol. 32, No. 3 (1994), pp. 207-210.
• Jean Coquet, Power sums of digital sums, J. Number Theory, Vol. 22, No. 2 (1986), pp. 161-176.
• 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.
• Robert E. Kennedy and Curtis N. Cooper, An extension of a theorem by Cheo and Yien concerning digital sums, Fibonacci Quarterly, Vol. 29, No. 2 (1991), pp. 145-149.
• 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.
• J.-L. Mauclaire and Leo Murata, On q-additive functions. II, Proc. Japan Acad. Ser. A Math. Sci., Vol. 59, No. 9 (1983), pp. 441-444.
• Harald Riede, Asymptotic estimation of a sum of digits, Fibonacci Quarterly, Vol. 36, No. 1 (1998), pp. 72-75.
• J. R. Trollope, An explicit expression for binary digital sums, Math. Mag., Vol. 41, No. 1 (1968), pp. 21-25.

Crossrefs

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

Partial sums of A118881.

Programs

Magma

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

Maple

See A037123.

Mathematica

Accumulate @ Array[(Plus @@ IntegerDigits[#])^2 &, 50] (* Amiram Eldar, Jan 20 2022 *)

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

Extensions

Offset changed to 0 and a(0) prepended by Amiram Eldar, Jan 20 2022