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A341915
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For any nonnegative number n with runs in binary expansion (r_1, ..., r_w), a(n) = Sum_{k = 1..w} 2^(r_1 + ... + r_k - 1).
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4
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0, 1, 3, 2, 5, 7, 6, 4, 9, 13, 15, 11, 10, 14, 12, 8, 17, 25, 29, 21, 23, 31, 27, 19, 18, 26, 30, 22, 20, 28, 24, 16, 33, 49, 57, 41, 45, 61, 53, 37, 39, 55, 63, 47, 43, 59, 51, 35, 34, 50, 58, 42, 46, 62, 54, 38, 36, 52, 60, 44, 40, 56, 48, 32, 65, 97, 113
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OFFSET
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0,3
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COMMENTS
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This sequence is a permutation of the nonnegative integers with inverse A341916.
This sequence has connections with A003188; here we compute partials sums of runs from left to right, there from right to left.
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LINKS
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FORMULA
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a(n) < 2^k for any n < 2^k.
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EXAMPLE
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For n = 23,
- the binary representation of 23 is "10111",
- the corresponding run lengths are (1, 1, 3),
- so a(23) = 2^(1-1) + 2^(1+1-1) + 2^(1+1+3-1) = 19.
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MATHEMATICA
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a[n_] := If[n == 0, 0, 2^((Length /@ Split[IntegerDigits[n, 2]] // Accumulate)-1) // Total];
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PROG
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(PARI) a(n) = { my (v=0); while (n, my (w=valuation(n+n%2, 2)); n\=2^w; v=2^w*(1+v)); v/2 }
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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