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.
A169947 = Import["https://oeis.org/A169947/b169947.txt", "Table"][[All, 2]]; Join[{1}, Differences[A169947]] (* Jean-François Alcover, Sep 02 2019 *)
b[n_, s_] := b[n, s] = Module[{sn, m}, m = Length[s]; sn = Append[s, n]; If[n < 1, 1, b[n - 1, s] + If[m*(m + 1)/2 == Length[Union[Flatten[Table[ sn[[i]] + sn[[j]], {i, 1, m}, {j, i + 1, m + 1}]]]], b[n - 1, sn], 0]]];
A196723[n_] := A196723[n] = b[n - 1, {n}] + If[n == 0, 0, A196723[n - 1]]; c[n_, s_] := c[n, s] = Module[{sn, m}, If[n < 1, 1, sn = Append[s, n]; m = Length[sn]; If[m*(m - 1)/2 == Length[Table[sn[[i]] - sn[[j]], {i, 1, m - 1}, {j, i + 1, m}] // Flatten // Union], c[n-1, sn], 0] + c[n-1, s]]];
A143823[n_] := A143823[n] = c[n - 1, {n}] + If[n == 0, 0, A143823[n - 1]];
a[n_] := A196723[n+1] - A196723[n] - A143823[n+1] + A143823[n];
Table[Print[n, " ", a[n]]; a[n], {n, 2, 42}] (* Jean-François Alcover, Sep 07 2019, after Alois P. Heinz in A196723 and A143823 *)
Triangle begins: [1] [1, 1] [1, 2, 1] [1, 1, 5, 1] [1, 2, 4, 8, 1] [1, 1, 8, 8, 13, 1] [1, 2, 7, 18, 15, 20, 1] [1, 1, 11, 13, 45, 23, 33, 1] [1, 2, 10, 28, 36, 86, 44, 48, 1] [1, 1, 14, 18, 84, 70, 184, 64, 75, 1] [1, 2, 13, 36, 68, 188, 166, 332, 117, 100, 1] [1, 1, 17, 23, 132, 134, 482, 282, 657, 173, 145, 1] [1, 2, 16, 44, 109, 316, 396, 1000, 601, 1144, 262, 204, 1] [1, 1, 20, 28, 187, 221, 924, 742, 2279, 1035, 2086, 374, 293, 1] [1, 2, 19, 52, 159, 478, 749, 2090, 1895, 4460, 2077, 3434, 571, 396, 1] [1, 1, 23, 33, 251, 327, 1561, 1469, 5403, 3463, 9441, 3397, 6047, 791, 559, 1] ...
For n = 3 we have the base-2 representation 11, the associated polynomial B_3(x) = x + 1, B_3(x)^2 = x^2 + 2*x + 1 and the magnitude of the largest coefficient in the expansion of B_3(x)^2 is 2, therefore a(3) = 2. For n = 4 we have the base-2 representation 100, the associated polynomial B_4(x) = x^2, B_4(x)^2 = x^4 and the magnitude of the largest coefficient in the expansion of B_4(x)^2 is 1, therefore a(4) = 1.
Table[Max@ CoefficientList[#, x] &[SeriesData[x, 0, #, 0, Length@ #, 1]^2] &@ Reverse@ IntegerDigits[n, 2], {n, 120}] (* Michael De Vlieger, Jun 08 2016 *)
a(n) = my(pol = Pol(binary(n))); return(vecmax(Vec(sqr(pol)))); concat(0, vector(100, n, a(n)))
bitrev(n) = subst(Pol(Vecrev(binary(n>>valuation(n,2))), 'x), 'x, 2);
a(n) = {
my(e = logint(n, 2), r = bitrev(n) << e, v = vector(2*e+1));
for (i = 1, #v, v[i] = hammingweight(bitand(r, n)); r >>= 1);
return(vecmax(v));
};
concat(0, vector(100, n, a(n)))
For n = 3, there are a(3) = 5 such polynomials:
f(x) | th(f(x)) | th(x*f(x)+1)
------------------+----------+-------------
1 + x + x^2 + x^3 | 4 | 5
1 + x^2 + x^3 | 2 | 4
1 + x + x^3 | 2 | 3
x + x^3 | 2 | 3
x^3 | 1 | 2
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