A231684 a(n) = Sum_{i=0..n} digsum_9(i), where digsum_9(i) = A053830(i).

0, 1, 3, 6, 10, 15, 21, 28, 36, 37, 39, 42, 46, 51, 57, 64, 72, 81, 83, 86, 90, 95, 101, 108, 116, 125, 135, 138, 142, 147, 153, 160, 168, 177, 187, 198, 202, 207, 213, 220, 228, 237, 247, 258, 270, 275, 281, 288, 296, 305, 315, 326, 338, 351, 357, 364, 372, 381, 391, 402, 414, 427, 441, 448, 456, 465, 475, 486, 498, 511, 525, 540, 548, 557, 567, 578, 590, 603, 617, 632

Offset: 0

Author

N. J. A. Sloane, Nov 13 2013

Keywords

• nonn
• base

References

• Jean-Paul Allouche and Jeffrey Shallit, Automatic sequences, Cambridge University Press, 2003, p. 94.

Links

• Amiram Eldar, Table of n, a(n) for n = 0..10000
• 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), pp. 263-271, Kluwer Acad. Publ., Dordrecht, 1993.
• Hsien-Kuei Hwang, Svante Janson and Tsung-Hsi Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, Vol. 13, No. 4 (2017), Article #47; ResearchGate link; preprint, 2016.
• 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.
• J. R. Trollope, An explicit expression for binary digital sums, Math. Mag., Vol. 41, No. 1 (1968), pp. 21-25.

Crossrefs

Cf. A053830, A231685, A231686, A231687.

Programs

Mathematica

a[n_] := Plus @@ IntegerDigits[n, 9]; Accumulate @ Array[a, 80, 0] (* Amiram Eldar, Dec 09 2021 *)

PARI

a(n) = sum(i=0, n, sumdigits(i, 9)); \\ Michel Marcus, Sep 20 2017

Formula

a(n) ~ 2*n*log(n)/log(3). - Amiram Eldar, Dec 09 2021