A231664 a(n) = Sum_{i=0..n} digsum_4(i), where digsum_4(i) = A053737(i).

0, 1, 3, 6, 7, 9, 12, 16, 18, 21, 25, 30, 33, 37, 42, 48, 49, 51, 54, 58, 60, 63, 67, 72, 75, 79, 84, 90, 94, 99, 105, 112, 114, 117, 121, 126, 129, 133, 138, 144, 148, 153, 159, 166, 171, 177, 184, 192, 195, 199, 204, 210, 214, 219, 225, 232, 237, 243, 250, 258, 264, 271, 279, 288, 289, 291, 294, 298, 300, 303, 307, 312, 315, 319, 324, 330, 334, 339, 345, 352, 354, 357

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

• Reinhard Zumkeller, 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. A053737, A231665, A231666, A231667.

Programs

Magma

[(&+[&+Intseq(j, 4): j in [0..n]]): n in [0..100]]; // G. C. Greubel, Feb 16 2019

Maple

ListTools:-PartialSums([seq(convert(convert(n,base,4),`+`), n=0..200)]); # Robert Israel, Sep 20 2017

Mathematica

Table[Sum[Total[IntegerDigits[j, 4]], {j,0,n}], {n, 0, 100}] (* G. C. Greubel, Feb 16 2019 *)

PARI

a(n) = sum(i=0, n, sumdigits(i, 4)); \\ Michel Marcus, Sep 20 2017

Formula

G.f.: g(x) satisfies g(x) = (1+x+x^2+x^3)^2*g(x^4) + (x+2*x^2+3*x^3)/(1-x-x^4+x^5). - Robert Israel, Sep 20 2017

a(n) ~ 3*n*log(n)/(4*log(2)). - Amiram Eldar, Dec 09 2021