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A083652
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Sum of lengths of binary expansions of 0 through n.
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18
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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
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OFFSET
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0,2
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COMMENTS
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Young writes "If n = 2^i + k [...] the maximum is (i+1)(2^i+k)-2^{i+1}+2." - Michael Somos, Sep 25 2012
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LINKS
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FORMULA
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a(n) = 2 + (n+1)*ceiling(log_2(n+1)) - 2^ceiling(log_2(n+1)).
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EXAMPLE
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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 + ...
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MATHEMATICA
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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 *)
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PROG
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(Haskell)
a083652 n = a083652_list !! n
a083652_list = scanl1 (+) a070939_list
(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)
s, i, z = 1, n, 1
while 0 <= i: s += i; i -= z; z += z
return s
(Python)
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CROSSREFS
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A296349 is an essentially identical sequence.
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KEYWORD
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nonn,easy,base
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AUTHOR
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STATUS
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approved
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