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

%I A048718 #25 Dec 31 2022 15:20:16
%S A048718 0,1,2,4,8,16,17,32,33,34,64,65,66,68,128,129,130,132,136,256,257,258,
%T A048718 260,264,272,273,512,513,514,516,520,528,529,544,545,546,1024,1025,
%U A048718 1026,1028,1032,1040,1041,1056,1057
%N A048718 Binary expansion matches ((0)*0001)*(0*); or, Zeckendorf-like expansion of n using recurrence f(n) = f(n-1) + f(n-4).
%C A048718 Max. 1 one-bit occur in each range of four bits.
%C A048718 Constructed from A003269 in the same way as A003714 is constructed from A000045.
%H A048718 Sebastian Karlsson, <a href="/A048718/a048718.txt">Walnut code that verifies the conjectures of Paul D. Hanna</a>
%H A048718 Walnut can be downloaded from <a href="https://cs.uwaterloo.ca/~shallit/walnut.html">https://cs.uwaterloo.ca/~shallit/walnut.html</a>.
%H A048718 <a href="/index/Con#CongruCrossDomain">Index entries for sequences defined by congruent products between domains N and GF(2)[X]</a>
%H A048718 <a href="/index/Con#CongruXOR">Index entries for sequences defined by congruent products under XOR</a>
%F A048718 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.
%F A048718 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
%F A048718 The conjectures can be verified using the Walnut theorem-prover (see links). - _Sebastian Karlsson_, Dec 31 2022
%t A048718 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] == {}];
%t A048718 Select[Range[0, 1057], filterQ] (* _Jean-François Alcover_, Dec 31 2020 *)
%o A048718 (PARI) is(n)=!bitand(n, 14*n) \\ _Charles R Greathouse IV_, Oct 03 2016
%Y A048718 Cf. A048715, A048719, A115422, A115423, A115424.
%K A048718 nonn,base,easy
%O A048718 0,3
%A A048718 _Antti Karttunen_, Mar 30 1999