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 A083652 Sum of lengths of binary expansions of 0 through n. 18
 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, and 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, and 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. Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 36. 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. Index entries for sequences related to binary expansion of n 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 (Python) def A083652(n): return 2+(n+1)*(m:=(n+1).bit_length())-(1<

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Last modified December 9 15:36 EST 2023. Contains 367693 sequences. (Running on oeis4.)