0,3

Max. 1 one-bit occur in each range of four bits.

Constructed from A003269 in the same way as A003714 is constructed from A000045.

Sebastian Karlsson, Walnut code that verifies the conjectures of Paul D. Hanna

Walnut can be downloaded from https://cs.uwaterloo.ca/~shallit/walnut.html.

a(0) = 0, a(n) = (2^(invfyy(n)-1))+a(n-fyy(invfyy(n))) where fyy(n) is fyy(n-1) + fyy(n-4) (A003269) and invfyy is its "integral" (floored down) inverse.

a(n) XOR 14*a(n) = 15*a(n); 3*a(n) XOR 9*a(n) = 10*a(n); 3*a(n) XOR 13*a(n) = 14*a(n); 5*a(n) XOR 9*a(n) = 12*a(n); 5*a(n) XOR 11*a(n) = 14*a(n); 6*a(n) XOR 11*a(n) = 13*a(n); 7*a(n) XOR 9*a(n) = 14*a(n); 7*a(n) XOR 10*a(n) = 13*a(n); 7*a(n) XOR 11*a(n) = 12*a(n); 12*a(n) XOR 21*a(n) = 25*a(n); 12*a(n) XOR 37*a(n) = 41*a(n); etc. (conjectures). - Paul D. Hanna, Jan 22 2006

The conjectures can be verified using the Walnut theorem-prover (see links). - Sebastian Karlsson, Dec 31 2022

filterQ[n_] := With[{bb = IntegerDigits[n, 2]}, !MemberQ[{{1, 1}, {1, 0, 1}, {1, 1, 0}, {1, 1, 1}}, bb] && SequencePosition[bb, {a_, b_, c_, d_} /; Count[{a, b, c, d}, 1] > 1] == {}];

Select[Range[0, 1057], filterQ] (* Jean-François Alcover, Dec 31 2020 *)

(PARI) is(n)=!bitand(n, 14*n) \\ Charles R Greathouse IV, Oct 03 2016

Cf. A048715, A048719, A115422, A115423, A115424.

Sequence in context: A061681 A100787 A115795 * A018510 A018366 A216781

Adjacent sequences: A048715 A048716 A048717 * A048719 A048720 A048721

nonn,base,easy

Antti Karttunen, Mar 30 1999

approved