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A355654 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)^2).

%I A355654 #6 Jul 14 2022 09:35:11
%S A355654 0,1,9,15,271,313,481,511,33279,34785,39993,40719,61455,61689,65409,
%T A355654 65535,16842751,17039233,17809657,17821711,20455183,20479033,20842465,
%U A355654 20939263,31457791,31465441,31584313,31588111,33488911,33489913,33553921,33554431,34393292799
%N A355654 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)^2).
%C A355654 In other words, a(n) encodes the square of the polynomial encoded by n.
%H A355654 Rémy Sigrist, <a href="/A355654/a355654.png">Binary plot of the sequence for n < 2^10</a>
%H A355654 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A355654 A005811(a(n)) = 2*A005811(n) - 1 for any n > 0.
%F A355654 a(2^n - 1) = 2^(n^2) - 1.
%e A355654 The first terms, alongside their binary expansions and corresponding polynomials, are:
%e A355654   n  a(n)   bin(n)  bin(a(n))         p(n)           p(a(n))
%e A355654   -  -----  ------  ----------------  -------------  -----------------------------
%e A355654   0      0       0                 0              0                              0
%e A355654   1      1       1                 1              1                              1
%e A355654   2      9      10              1001          x + 1                  x^2 + 2*x + 1
%e A355654   3     15      11              1111              2                              4
%e A355654   4    271     100         100001111          x + 2                  x^2 + 4*x + 4
%e A355654   5    313     101         100111001    x^2 + x + 1  x^4 + 2*x^3 + 3*x^2 + 2*x + 1
%e A355654   6    481     110         111100001        2*x + 1                4*x^2 + 4*x + 1
%e A355654   7    511     111         111111111              3                              9
%e A355654   8  33279    1000  1000000111111111          x + 3                  x^2 + 6*x + 9
%e A355654   9  34785    1001  1000011111100001  x^2 + 2*x + 1  x^4 + 4*x^3 + 6*x^2 + 4*x + 1
%o A355654 (PARI) toruns(n) = { my (r=[]); while (n, my (v=valuation(n+n%2, 2)); n\=2^v; r=concat(v, r)); r }
%o A355654 fromruns(r) = { my (v=0); for (k=1, #r, v=(v+k%2)*2^r[k]-k%2); v }
%o A355654 a(n) = { fromruns(Vec(Pol(toruns(n))^2)) }
%Y A355654 Cf. A005811, A101211, A212739, A355653.
%K A355654 nonn,base
%O A355654 0,3
%A A355654 _Rémy Sigrist_, Jul 12 2022