A115423 -id:A115423 - cp's OEIS frontend

A048715 Binary expansion matches (100(0))(0|1|10)?; or, Zeckendorf-like expansion of n using recurrence f(n) = f(n-1) + f(n-3).

0, 1, 2, 4, 8, 9, 16, 17, 18, 32, 33, 34, 36, 64, 65, 66, 68, 72, 73, 128, 129, 130, 132, 136, 137, 144, 145, 146, 256, 257, 258, 260, 264, 265, 272, 273, 274, 288, 289, 290, 292, 512, 513, 514, 516, 520, 521, 528, 529, 530, 544, 545, 546, 548, 576, 577, 578, 580

Offset: 0

Antti Karttunen, Mar 30 1999

No more than one 1-bit in each bit triple.
All terms satisfy A048727(n) = 7*n.
Constructed from A000930 in the same way as A003714 is constructed from A000045.
It appears that n is in the sequence if and only if C(7n,n) is odd (cf. A003714). - Benoit Cloitre, Mar 09 2003
The conjecture by Benoit is correct. This is easily proved using the well-known result that the multiplicity with which a prime p divides C(n+m,n) is the number of carries when adding n+m in base p. - Franklin T. Adams-Watters, Oct 06 2009
Appears to be the set of numbers x such that (x AND 5*x) = x and (x OR 3*x)/x = 3. - Gary Detlefs, Jun 08 2024

G. C. Greubel, Table of n, a(n) for n = 0..1275
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
Index entries for 2-automatic sequences.

Subsequence of A048716.

Cf. A003726, A004742, A004743, A004744, A048717, A048718, A048719, A048730, A048733, A115422, A115423, A115424.

Reap[Do[If[OddQ[Binomial[7n, n]], Sow[n]], {n, 0, 400}]][[2, 1]]
(* Second program: *)
filterQ[n_] := With[{bb = IntegerDigits[n, 2]}, !MatchQ[bb, {_, 1, 0, 1, _}|{_, 1, 1, _}]];
Select[Range[0, 580], filterQ] (* Jean-François Alcover, Dec 31 2020 *)

is(n)=!bitand(n, 6*n) \\ Charles R Greathouse IV, Oct 03 2016

for my $k (0..580) { print "$k, " if sprintf("%b", $k) =~ m{^(100(0)*)*(0|1|10)?$}; } # Georg Fischer, Jun 26 2021

import re
def ok(n): return re.fullmatch('(100(0)*)*(0|1|10)?', bin(n)[2:]) != None
print(list(filter(ok, range(581)))) # Michael S. Branicky, Jun 26 2021

a(0) = 0, a(n) = (2^(invfoo(n)-1))+a(n-foo(invfoo(n))), where foo(n) is foo(n-1) + foo(n-3) (A000930) and invfoo is its "integral" (floored down) inverse.
a(n) XOR 6*a(n) = 7*a(n); 3*a(n) XOR 4*a(n) = 7*a(n); 3*a(n) XOR 5*a(n) = 6*a(n); (conjectures). - Paul D. Hanna, Jan 22 2006
The conjectures can be verified using the Walnut theorem-prover (see links). - Sebastian Karlsson, Dec 31 2022

Definition corrected by Georg Fischer, Jun 26 2021

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

Original entry on oeis.org

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

Antti Karttunen, Mar 30 1999

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.
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

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

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 *)

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

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

A115424 Integers n > 0 such that n XOR 62n = 63n.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 65, 128, 129, 130, 256, 257, 258, 260, 512, 513, 514, 516, 520, 1024, 1025, 1026, 1028, 1032, 1040, 2048, 2049, 2050, 2052, 2056, 2064, 2080, 4096, 4097, 4098, 4100, 4104, 4112, 4128, 4160, 4161, 8192, 8193, 8194, 8196, 8200, 8208

Offset: 1

Paul D. Hanna, Jan 22 2006

nonn

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A003714 (Fibbinary numbers), A048715, A048718, A115422, A115423.

Select[Range[8300],BitXor[#,62 #]==63 #&] (* Harvey P. Dale, Oct 31 2021 *)

isok(n) = bitxor(n, 62*n) == 63*n; \\ Michel Marcus, Nov 11 2016

$cnt=0;
foreach(1..8_000){
    print ++$cnt," $\n" if ((62*$)^$)==63*$;
}
# Ivan Neretin, Nov 11 2016

This sequence also seems to satisfy:
3*a(n) XOR 41*a(n) = 42*a(n);
5*a(n) XOR 35*a(n) = 38*a(n);
6*a(n) XOR 35*a(n) = 37*a(n);
7*a(n) XOR 35*a(n) = 36*a(n); etc.

A115422 Integers n > 0 such that n XOR 20n = 21n.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 33, 36, 48, 64, 65, 66, 67, 72, 73, 96, 97, 128, 129, 130, 131, 132, 134, 144, 146, 192, 193, 194, 195, 256, 257, 258, 259, 260, 262, 264, 265, 268, 288, 289, 292, 384, 385, 386, 387, 388, 390, 512, 513, 514, 515, 516, 518

Offset: 1

Paul D. Hanna, Jan 22 2006

nonn

n is in the sequence iff 2*n is in the sequence. - Robert Israel, Nov 11 2016

Ivan Neretin, Table of n, a(n) for n = 1..10203

Cf. A003714 (Fibbinary numbers), A048715, A048718, A115423, A115424.

select(n -> Bits:-Xor(n,20*n)=21*n, [$1..1000]); # Robert Israel, Nov 11 2016

Select[Range[600],BitXor[#,20#]==21#&] (* Harvey P. Dale, Apr 21 2018 *)

isok(n) = bitxor(n, 20*n) == 21*n; \\ Michel Marcus, Nov 11 2016

$cnt=0;
foreach(1..1_000){
    print ++$cnt," $\n" if ((20*$)^$)==21*$;
}
# Ivan Neretin, Nov 10 2016

This sequence also seems to satisfy:
5*a(n) XOR 16*a(n) = 21*a(n);
5*a(n) XOR 17*a(n) = 20*a(n); etc.
a(A224809(n+4)) = 2^n. - Gheorghe Coserea, Nov 11 2016

cp's OEIS Frontend

A048715 Binary expansion matches (100(0)*)*(0|1|10)?; or, Zeckendorf-like expansion of n using recurrence f(n) = f(n-1) + f(n-3).

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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).

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A115424 Integers n > 0 such that n XOR 62*n = 63*n.

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A115422 Integers n > 0 such that n XOR 20*n = 21*n.

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A048715 Binary expansion matches (100(0))(0|1|10)?; or, Zeckendorf-like expansion of n using recurrence f(n) = f(n-1) + f(n-3).

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

A115424 Integers n > 0 such that n XOR 62n = 63n.

A115422 Integers n > 0 such that n XOR 20n = 21n.