A231501 a(n) = Sum_{i=0..n} wt(i)^3, where wt() = A000120().

0, 1, 2, 10, 11, 19, 27, 54, 55, 63, 71, 98, 106, 133, 160, 224, 225, 233, 241, 268, 276, 303, 330, 394, 402, 429, 456, 520, 547, 611, 675, 800, 801, 809, 817, 844, 852, 879, 906, 970, 978, 1005, 1032, 1096, 1123, 1187, 1251, 1376, 1384, 1411, 1438, 1502, 1529, 1593, 1657, 1782, 1809, 1873, 1937, 2062, 2126, 2251

Offset

0,3

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), Kluwer Acad. Publ., Dordrecht, 1993, pp. 263-271; alternative link.

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.

Kenneth B. Stolarsky, Power and exponential sums of digital sums related to binomial coefficient parity, SIAM J. Appl. Math., Vol. 32, No. 4 (1977), pp. 717-730.

J. R. Trollope, An explicit expression for binary digital sums, Math. Mag., Vol. 41, No. 1 (1968), pp. 21-25.

Formula

a(n) ~ n * (log(n)/(2*log(2)))^3 + O(n*log(n)^2) (Stolarsky, 1977). - Amiram Eldar, Jan 20 2022

a(n) = Sum_{k=0..floor(log_2(n+1))} k^3 * A360189(n,k). - Alois P. Heinz, Mar 06 2023

Mathematica

Accumulate @ (Table[DigitCount[n, 2, 1], {n, 0, 60}]^3) (* Amiram Eldar, Jan 20 2022 *)

Prog

(PARI) a(n) = sum(i=0, n, hammingweight(i)^3); \\ Michel Marcus, Sep 20 2017

Crossrefs

Cf. A000120, A000788, A231500, A231502, A360189.

Sequence in context: A282093 A032930 A033293 * A051373 A131096 A069263

Adjacent sequences: A231498 A231499 A231500 * A231502 A231503 A231504

Keyword

nonn,base

Author

N. J. A. Sloane, Nov 12 2013

Status

approved