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
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.
Programs
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Mathematica
a[n_] := Plus @@ IntegerDigits[n, 3]; Accumulate @ Array[a, 60, 0] (* Amiram Eldar, Dec 09 2021 *)
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PARI
s(k,n)=n-(k-1)*sum(m=1,n,valuation(m,k)); a(n)=sum(i=0,n,s(3,i))
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PARI
a(n)= sum(i=1, n, sumdigits(i, 3)); \\ Ruud H.G. van Tol, Nov 19 2024
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).