A048719 -id:A048719 - 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

A048716 Numbers n such that binary expansion matches ((0)00(1?)1)(0*).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 9, 12, 16, 17, 18, 19, 24, 25, 32, 33, 34, 35, 36, 38, 48, 49, 50, 51, 64, 65, 66, 67, 68, 70, 72, 73, 76, 96, 97, 98, 99, 100, 102, 128, 129, 130, 131, 132, 134, 136, 137, 140, 144, 145, 146, 147, 152, 153, 192, 193, 194, 195, 196, 198, 200, 201

Offset: 1

Antti Karttunen, Mar 30 1999

If bit i is 1, then bits i+-2 must be 0. All terms satisfy A048725(n) = 5*n.
It appears that n is in the sequence if and only if C(5n,n) is odd (cf. A003714). - Benoit Cloitre, Mar 09 2003
Yes, as remarked in A048715, "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." - Jason Kimberley, Dec 21 2011
A116361(a(n)) <= 2. - Reinhard Zumkeller, Feb 04 2006

Superset of A048715 and A048719. Union of A004742 and A003726.

Cf. A048729, A003714, A115845, A115847, A116360.

Reap[Do[If[OddQ[Binomial[5n, n]], Sow[n]], {n, 0, 400}]][[2, 1]]
(* Second program: *)
filterQ[n_] := With[{bb = IntegerDigits[n, 2]}, MatchQ[bb, {0}|{1}|{1, 1}|{_, 0, , 1, __}|{_ 1, , 0, __}] && !MatchQ[bb, {_, 1, , 1, __}]];
Select[Range[0, 201], filterQ] (* Jean-François Alcover, Dec 31 2020 *)

is(n)=!bitand(n,n>>2) \\ Charles R Greathouse IV, Oct 03 2016

list(lim)=my(v=List(),n,t); while(n<=lim, t=bitand(n,n>>2); if(t, n+=1<Charles R Greathouse IV, Oct 22 2021

A048717 Binary expansion matches ((0)00(1)11)(0).

Original entry on oeis.org

0, 3, 6, 7, 12, 14, 15, 24, 28, 30, 31, 48, 51, 56, 60, 62, 63, 96, 99, 102, 103, 112, 115, 120, 124, 126, 127, 192, 195, 198, 199, 204, 206, 207, 224, 227, 230, 231, 240, 243, 248, 252, 254, 255, 384, 387, 390, 391

Offset: 0

Antti Karttunen, Mar 30 1999

In binary expansion, 1-bits occur only in groups of two or more, separated from other such groups by at least two 0-bits.

Integers that satisfy A048727(n) = 3*n.

Index entries for sequences defined by congruent products between domains N and GF(2)[X]

Row 3 of A115872. Superset of A048719. Cf. A048733.

filterQ[n_] := !MatchQ[IntegerDigits[n, 2], {1}|{1, 0, _}|{_, 0, 1}|{_, 1, 0, 1, _}|{_, 0, 1, 0, _}];
Select[Range[0, 400], filterQ] (* Jean-François Alcover, Dec 31 2020 *)

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

A115767 Integers i such that 2i XOR 5i = 3*i.

Original entry on oeis.org

0, 3, 6, 11, 12, 22, 24, 43, 44, 48, 51, 86, 88, 96, 99, 102, 171, 172, 176, 179, 192, 195, 198, 203, 204, 342, 344, 352, 355, 358, 384, 387, 390, 395, 396, 406, 408, 683, 684, 688, 691, 704, 707, 710, 715, 716, 768, 771, 774, 779, 780, 790, 792, 811, 812

Offset: 0

Antti Karttunen, Jan 30 2006

nonn

XOR is A003987.

Index entries for sequences defined by congruent products under XOR

Superset of A048719. A115768 gives the (?) terms which do not occur in A048719. Cf. A115424.

Select[Range[0,900],BitXor[2#,5#]==3#&] (* Harvey P. Dale, Jul 28 2017 *)

is(n)=bitxor(2*n, 5*n)==3*n \\ Charles R Greathouse IV, Sep 25 2024

A115768 Integers i such that 2i XOR 5i = 3i, but 4i XOR i is not 5*i.

Original entry on oeis.org

11, 22, 43, 44, 86, 88, 171, 172, 176, 179, 203, 342, 344, 352, 355, 358, 395, 406, 683, 684, 688, 691, 704, 707, 710, 715, 716, 779, 790, 811, 812, 1366, 1368, 1376, 1379, 1382, 1408, 1411, 1414, 1419, 1420, 1430, 1432, 1547, 1558, 1579, 1580

Offset: 1

Antti Karttunen, Jan 30 2006

nonn

XOR is A003987.

Cf. Setwise difference of A115767 and A048719 ? A115769 shows this sequence in binary.

A115807 Integers i such that 15*i = 27 X i.

Original entry on oeis.org

0, 3, 6, 12, 24, 48, 51, 96, 102, 192, 195, 204, 384, 387, 390, 408, 768, 771, 774, 780, 816, 819, 1536, 1539, 1542, 1548, 1560, 1632, 1638, 3072, 3075, 3078, 3084, 3096, 3120, 3123, 3264, 3267, 3276, 6144, 6147, 6150, 6156, 6168, 6192, 6195, 6240

Offset: 0

Antti Karttunen, Feb 01 2006

nonn

Here * stands for ordinary multiplication and X means carryless (GF(2)[X]) multiplication (A048720).

Index entries for sequences defined by congruent products between domains N and GF(2)[X]

Cf. Subset of A048719 ? Differs from A048719 for the first time at n=8, where A048719(8)=99, while a(8)=102. (Note: 99 = 1100011 in binary). A115808 shows this sequence in binary.

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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A048716 Numbers n such that binary expansion matches ((0)*00(1?)1)*(0*).

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A048717 Binary expansion matches ((0)*00(1*)11)*(0*).

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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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A115767 Integers i such that 2*i XOR 5*i = 3*i.

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A115768 Integers i such that 2*i XOR 5*i = 3*i, but 4*i XOR i is not 5*i.

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A115807 Integers i such that 15*i = 27 X i.

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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).

A048716 Numbers n such that binary expansion matches ((0)00(1?)1)(0*).

A048717 Binary expansion matches ((0)00(1)11)(0).

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

A115767 Integers i such that 2i XOR 5i = 3*i.

A115768 Integers i such that 2i XOR 5i = 3i, but 4i XOR i is not 5*i.