A094345 Sum of all digits in ternary expansions of 0, ..., n.

0, 1, 3, 4, 6, 9, 11, 14, 18, 19, 21, 24, 26, 29, 33, 36, 40, 45, 47, 50, 54, 57, 61, 66, 70, 75, 81, 82, 84, 87, 89, 92, 96, 99, 103, 108, 110, 113, 117, 120, 124, 129, 133, 138, 144, 147, 151, 156, 160, 165, 171, 176, 182, 189, 191, 194, 198, 201, 205, 210, 214, 219

Offset

0,3

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.

Formula

Asymptotically: a(n) = n*log(n)/log(3) + n*F(log(n)/log(3)) where F is a continuous function of period 1 nowhere differentiable (see Allouche & Shallit book).

Mathematica

a[n_] := Plus @@ IntegerDigits[n, 3]; Accumulate @ Array[a, 60, 0] (* Amiram Eldar, Dec 09 2021 *)

Prog

(PARI) s(k, n)=n-(k-1)*sum(m=1, n, valuation(m, k));
a(n)=sum(i=0, n, s(3, i))

Crossrefs

Cf. A000788, A053735, A231503, A231504, A231505.

Sequence in context: A184795 A285412 A105527 * A243302 A301654 A289233

Adjacent sequences: A094342 A094343 A094344 * A094346 A094347 A094348

Keyword

nonn,base

Author

Benoit Cloitre, Jun 08 2004

Status

approved