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 A083652 Sum of lengths of binary expansions of 0 through n. 15
 1, 2, 4, 6, 9, 12, 15, 18, 22, 26, 30, 34, 38, 42, 46, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130, 136, 142, 148, 154, 160, 166, 172, 178, 184, 190, 196, 202, 208, 214, 220, 226, 232, 238, 244, 250, 256, 262, 268, 274, 280, 286, 292 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS a(n) = A001855(n) + 1 for n > 0; a(0) = A070939(0)=1, n > 0: a(n) = a(n-1) + A070939(n). A030190(a(k))=1; A030530(a(k)) = k + 1. Partial sums of A070939. - Hieronymus Fischer, Jun 12 2012 Young writes "If n = 2^i + k [...] the maximum is (i+1)(2^i+k)-2^{i+1}+2." - Michael Somos, Sep 25 2012 LINKS Reinhard Zumkeller, Table of n, a(n) for n = 0..10000 Hsien-Kuei Hwang, S. Janson, T.-H. Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint 2016. Hsien-Kuei Hwang, S. Janson, T.-H. Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585. Alfred Young, The Maximum Order of an Irreducible Covariant of a System of Binary Forms, Proc. Roy. Soc. 72 (1903), 399-400 = The Collected Papers of Alfred Young, 1977, 136-137. FORMULA a(n) = 2 + (n+1)*ceiling(log_2(n+1)) - 2^ceiling(log_2(n+1)). G.f.: g(x) = 1/(1-x) + (1/(1-x)^2)*Sum_{j>=0} x^2^j. - Hieronymus Fischer, Jun 12 2012; corrected by Ilya Gutkovskiy, Jan 08 2017 a(n) = A123753(n) - n. - Peter Luschny, Nov 30 2017 EXAMPLE G.f. = 1 + 2*x + 4*x^2 + 6*x^3 + 9*x^4 + 12*x^5 + 15*x^6 + 18*x^7 + 22*x^8 + ... MATHEMATICA Accumulate[Length/@(IntegerDigits[#, 2]&/@Range[0, 60])] (* Harvey P. Dale, May 28 2013 *) a[n_] := (n + 1) IntegerLength[n + 1, 2] - 2^IntegerLength[n + 1, 2] + 2; Table[a[n], {n, 0, 58}] (* Peter Luschny, Dec 02 2017 *) PROG (Haskell) a083652 n = a083652_list !! n a083652_list = scanl1 (+) a070939_list -- Reinhard Zumkeller, Jul 05 2012 (PARI) {a(n) = my(i); if( n<0, 0, n++; i = length(binary(n)); n*i - 2^i + 2)}; /* Michael Somos, Sep 25 2012 */ (PARI) a(n)=my(i=#binary(n++)); n*i-2^i+2 \\ equivalent to the above (Python) def A083652(n):     s, i, z = 1, n, 1     while 0 <= i: s += i; i -= z; z += z     return s print([A083652(n) for n in range(0, 59)]) # Peter Luschny, Nov 30 2017 CROSSREFS Cf. A000120, A007088, A023416, A059015, A070939 (base 2), A123753. A296349 is an essentially identical sequence. Sequence in context: A130240 A143118 A162800 * A296349 A325544 A118103 Adjacent sequences:  A083649 A083650 A083651 * A083653 A083654 A083655 KEYWORD nonn,easy,base AUTHOR Reinhard Zumkeller, May 01 2003 STATUS approved

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Last modified October 19 21:28 EDT 2019. Contains 328244 sequences. (Running on oeis4.)