A207974 -id:A207974 - cp's OEIS frontend

0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 2, 2, 2, 3, 4, 1, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 4, 3, 3, 4, 5, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 3, 2, 4, 3, 4, 2, 2, 2, 4, 2, 3, 4, 4, 3, 3, 3, 4, 4, 4, 5, 6, 1, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 3, 3, 4, 2, 2, 2, 2, 3, 4, 3, 4, 2, 3, 4, 4, 3, 5, 4, 5, 2, 2, 2, 4, 2

Offset: 0

Gheorghe Coserea, Jun 07 2016

Let b_k..b_0 be the binary representation of n and B_n(x) = b_k*x^k + .. + b_0 the associated polynomial with n = B_n(2); we define the thickness of n to be the thickness of B_n, i.e., the magnitude of the largest coefficient in the expansion of B_n(x)^2 (see A169950).
The thickness histogram for numbers in the interval I_n = [2^n, 2^(n+1)-1] is given by row n of triangle A169950, i.e., A169950(n,k) = card {p, p in I_n and a(p) = k}.
In general a(n) <= A000120(n), with equality only if in base-2 n becomes a palindrome after trailing 0's (if any) are omitted, i.e., n = A057890(k) for some k; the number of such numbers in I_n having binary weight (and thickness) w is given by A207974(n,w-1), i.e., A207974(n,w-1) = card {k, A057890(k) in I_n and A000120(A057890(k)) = w}; the total number of these numbers in the interval I_n is given by A027383(n), i.e., card {p, p in I_n and a(p) = A000120(p)} = A027383(n) = 2^floor((n+2)/2) + 2^floor((n+1)/2) - 2.

			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.

Gheorghe Coserea, Table of n, a(n) for n = 0..32768

Cf. A000120, A027383, A057890, A169950, A207974.

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)))

a(n) <= A000120(n), with equality iff n = A057890(k).

cp's OEIS Frontend

A274036 a(n) is the thickness of n (see Comments section for definition).

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