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A355653
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For any number n with runs in binary expansion (r_w, ..., r_0), let p(n) be the polynomial of a single indeterminate x where the coefficient of x^e is r_e for e = 0..w and otherwise 0, and let q be the inverse of p; a(n) = q(p(n)').
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2
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0, 0, 1, 0, 1, 6, 3, 0, 1, 12, 57, 6, 3, 30, 7, 0, 1, 24, 225, 12, 57, 966, 115, 6, 3, 60, 505, 30, 7, 126, 15, 0, 1, 48, 897, 24, 225, 7686, 451, 12, 57, 1932, 31801, 966, 115, 3870, 231, 6, 3, 120, 2017, 60, 505, 16326, 1011, 30, 7, 252, 4089, 126, 15, 510
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OFFSET
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0,6
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COMMENTS
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In other words, a(n) encodes the derivative of the polynomial encoded by n.
For n > 0, the runs in binary expansion of a(n) are (w*r_w, ..., 1*r_1).
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LINKS
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FORMULA
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a^A005811(n)(n) = 0 (where a^k denotes the k-th iterate of a).
a(n) = 1 iff n = 2^k for some k > 0.
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EXAMPLE
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The first terms, alongside their binary expansions and corresponding polynomials, are:
n a(n) bin(n) bin(a(n)) p(n) p(a(n))
-- ---- ------ --------- ----------------- ---------------
0 0 0 0 0 0
1 0 1 0 1 0
2 1 10 1 x + 1 1
3 0 11 0 2 0
4 1 100 1 x + 2 1
5 6 101 110 x^2 + x + 1 2*x + 1
6 3 110 11 2*x + 1 2
7 0 111 0 3 0
8 1 1000 1 x + 3 1
9 12 1001 1100 x^2 + 2*x + 1 2*x + 2
10 57 1010 111001 x^3 + x^2 + x + 1 3*x^2 + 2*x + 1
11 6 1011 110 x^2 + x + 2 2*x + 1
12 3 1100 11 2*x + 2 2
13 30 1101 11110 2*x^2 + x + 1 4*x + 1
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PROG
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(PARI) toruns(n) = { my (r=[]); while (n, my (v=valuation(n+n%2, 2)); n\=2^v; r=concat(v, r)); r }
fromruns(r) = { my (v=0); for (k=1, #r, v=(v+k%2)*2^r[k]-k%2); v }
a(n) = { fromruns(Vec(deriv(Pol(toruns(n))))) }
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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