A048718 Binary expansion matches ((0)*0001)*(0*); or, Zeckendorf-like expansion of n using recurrence f(n) = f(n-1) + f(n-4).

0, 1, 2, 4, 8, 16, 17, 32, 33, 34, 64, 65, 66, 68, 128, 129, 130, 132, 136, 256, 257, 258, 260, 264, 272, 273, 512, 513, 514, 516, 520, 528, 529, 544, 545, 546, 1024, 1025, 1026, 1028, 1032, 1040, 1041, 1056, 1057

Offset: 0

Author

Antti Karttunen, Mar 30 1999

Keywords

• nonn
• base
• easy

Comments

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

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

Links

• Sebastian Karlsson, Walnut code that verifies the conjectures of Paul D. Hanna
• Walnut can be downloaded from https://cs.uwaterloo.ca/~shallit/walnut.html.
• Index entries for sequences defined by congruent products between domains N and GF(2)[X]
• Index entries for sequences defined by congruent products under XOR

Crossrefs

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

Programs

Mathematica

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

Formula

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