This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A355653 #12 Jul 14 2022 15:02:01
%S A355653 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,
%T A355653 30,7,126,15,0,1,48,897,24,225,7686,451,12,57,1932,31801,966,115,3870,
%U A355653 231,6,3,120,2017,60,505,16326,1011,30,7,252,4089,126,15,510
%N A355653 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)').
%C A355653 In other words, a(n) encodes the derivative of the polynomial encoded by n.
%C A355653 For n > 0, the runs in binary expansion of a(n) are (w*r_w, ..., 1*r_1).
%H A355653 Rémy Sigrist, <a href="/A355653/b355653.txt">Table of n, a(n) for n = 0..8192</a>
%H A355653 Rémy Sigrist, <a href="/A355653/a355653.png">Binary plot of the sequence for n = 0..2^10</a>
%H A355653 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A355653 a^A005811(n)(n) = 0 (where a^k denotes the k-th iterate of a).
%F A355653 a(n) = 0 iff n belongs to A000225.
%F A355653 a(n) = 1 iff n = 2^k for some k > 0.
%F A355653 A005811(a(n)) = A005811(n)-1 for any n > 0.
%e A355653 The first terms, alongside their binary expansions and corresponding polynomials, are:
%e A355653 n a(n) bin(n) bin(a(n)) p(n) p(a(n))
%e A355653 -- ---- ------ --------- ----------------- ---------------
%e A355653 0 0 0 0 0 0
%e A355653 1 0 1 0 1 0
%e A355653 2 1 10 1 x + 1 1
%e A355653 3 0 11 0 2 0
%e A355653 4 1 100 1 x + 2 1
%e A355653 5 6 101 110 x^2 + x + 1 2*x + 1
%e A355653 6 3 110 11 2*x + 1 2
%e A355653 7 0 111 0 3 0
%e A355653 8 1 1000 1 x + 3 1
%e A355653 9 12 1001 1100 x^2 + 2*x + 1 2*x + 2
%e A355653 10 57 1010 111001 x^3 + x^2 + x + 1 3*x^2 + 2*x + 1
%e A355653 11 6 1011 110 x^2 + x + 2 2*x + 1
%e A355653 12 3 1100 11 2*x + 2 2
%e A355653 13 30 1101 11110 2*x^2 + x + 1 4*x + 1
%o A355653 (PARI) toruns(n) = { my (r=[]); while (n, my (v=valuation(n+n%2, 2)); n\=2^v; r=concat(v, r)); r }
%o A355653 fromruns(r) = { my (v=0); for (k=1, #r, v=(v+k%2)*2^r[k]-k%2); v }
%o A355653 a(n) = { fromruns(Vec(deriv(Pol(toruns(n))))) }
%Y A355653 Cf. A000225, A005811, A101211, A355654.
%K A355653 nonn,base
%O A355653 0,6
%A A355653 _Rémy Sigrist_, Jul 12 2022