A161674 -id:A161674 - cp's OEIS frontend

A161817 Positions n such that A010060(n) = A010060(n+5).

0, 2, 5, 8, 10, 11, 12, 14, 15, 16, 18, 21, 24, 26, 29, 32, 34, 37, 40, 42, 43, 44, 46, 47, 48, 50, 53, 56, 58, 59, 60, 62, 63, 64, 66, 69, 72, 74, 75, 76, 78, 79, 80, 82, 85, 88, 90, 93, 96, 98, 101, 104, 106, 107, 108, 110, 111, 112, 114, 117, 120, 122, 125, 128, 130, 133, 136, 138, 139, 140, 142, 143, 144

Offset: 1

Vladimir Shevelev, Jun 20 2009

Let A=Axxxxxx be any sequence. Denote by A^* the intersection of A and the union of sequences {4*A(n)+k}, k=-1,0,1,2. Then the present sequence is the union of A079523^* and A121539^*.

Conjecture. In every sequence of numbers n such that A010060(n)=A010060(n+k) for fixed odd k, the odious (A000069) and evil (A001969) terms alternate. [Vladimir Shevelev, Jul 31 2009]

G. C. Greubel, Table of n, a(n) for n = 1..10000
J.-P. Allouche, Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence, arXiv:1401.3727 [math>NT], 2014.
J.-P. Allouche, Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence, J. de Théorie des Nombres de Bordeaux, 27, no. 2 (2015), 375-388.
V. Shevelev, Equations of the form t(x+a)=t(x) and t(x+a)=1-t(x) for Thue-Morse sequence, arXiv:0907.0880 [math.NT], 2009-2012.

Cf. A161674, A161673, A161639, A161641, A161627, A161579, A161580, A121539, A131323, A036554, A010060, A079523, A081706.

tm[0] = 0; tm[n_?EvenQ] := tm[n] = tm[n/2]; tm[n_] := tm[n] = 1 - tm[(n - 1)/2]; Reap[For[n = 0, n <= 20000, n++,  If[tm[n] == tm[n + 5], Sow[n]]]][[2, 1]] (* G. C. Greubel, Jan 05 2018 *)

is(n)=hammingweight(n+5)==Mod(hammingweight(n),2) \\ Charles R Greathouse IV, Mar 26 2013

A161824 Numbers such that A010060(n) = A010060(n+6).

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 8, 9, 12, 13, 16, 17, 18, 19, 22, 23, 24, 25, 26, 27, 30, 31, 32, 33, 34, 35, 38, 39, 40, 41, 44, 45, 48, 49, 50, 51, 54, 55, 56, 57, 60, 61, 64, 65, 66, 67, 70, 71, 72, 73, 76, 77, 80, 81, 82, 83, 86, 87, 88, 89, 90, 91, 94, 95, 96, 97, 98, 99, 102, 103, 104, 105, 108

Offset: 1

Vladimir Shevelev, Jun 20 2009

Let A=Axxxxxx be any sequence from OEIS. Denote by A^* the intersection of the union of sequences {2*A(n)+j}, j=0,1, and the union of sequences {4*A(n)+k}, k=-2,-1,0,1. Then the sequence is the union of (A079523)^* and (A121539)^*.

G. C. Greubel, Table of n, a(n) for n = 1..10000
J.-P. Allouche, Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence, arXiv:1401.3727 [math.NT], 2014.
J.-P. Allouche, Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence, J. de Théorie des Nombres de Bordeaux, 27, no. 2 (2015), 375-388.
Vladimir Shevelev, Equations of the form t(x+a)=t(x) and t(x+a)=1-t(x) for Thue-Morse sequence, arXiv:0907.0880 [math.NT], 2009-2012.

Cf. A161817, A161674, A161673, A161639, A161641, A161627, A161579, A161580, A121539, A131323, A036554, A010060, A079523, A081706.

tm[0] = 0; tm[n_?EvenQ] := tm[n] = tm[n/2]; tm[n_] := tm[n] = 1 - tm[(n - 1)/2]; Reap[For[n = 0, n <= 6000, n++, If[tm[n] == tm[n + 6], Sow[n]]]][[2, 1]] (* G. C. Greubel, Jan 05 2018 *)

is(n)=hammingweight(n)%2==hammingweight(n+6)%2 \\ Charles R Greathouse IV, Aug 20 2013

Terms a(40) onwards added by G. C. Greubel, Jan 05 2018

Offset corrected by Mohammed Yaseen, Mar 29 2023

A161890 Numbers such that A010060(n) = A010060(n+9).

Original entry on oeis.org

0, 2, 3, 4, 6, 7, 9, 13, 15, 16, 18, 19, 20, 22, 24, 26, 27, 28, 30, 32, 34, 35, 36, 38, 39, 41, 45, 47, 48, 50, 51, 52, 54, 55, 57, 61, 63, 64, 66, 67, 68, 70, 71, 73, 77, 79, 80, 82, 83, 84, 86, 88, 90, 91, 92, 94, 96, 98, 99, 100, 102, 103, 105, 109, 111, 112, 114, 115, 116, 118, 120

Offset: 0

Vladimir Shevelev, Jun 21 2009

Or union of intersection of A161639 and {A079523(n)-8} and intersection of A161673 and {A121539(n)-8}. In general, for a>=1, consider equations A010060(x+a)+A010060(x)=1, A010060(x+a)=A010060(x). Denote via B_a (C_a) the sequence of nonnegative solutions of the first (second) equation. Then we have recursions: B_(a+1) is the union of transactions 1) C_a and {A121539(n)-a}, 2) B_a and {A079523(n)-a}; C_(a+1) is the union of transactions 1) C_a and {A079523(n)-a}, 2) B_a and {A121539(n)-a}.
Conjecture. In every sequence of numbers n, such that A010060(n)=A010060(n+k), for fixed odd k, the odious (A000069) and evil (A001969) terms alternate. - Vladimir Shevelev, Jul 31 2009
This conjecture was actually proved in a later version of the Shevelev arxiv article cited below, and it can also easily be proved by the Walnut prover. - Jeffrey Shallit, Oct 12 2022

G. C. Greubel, Table of n, a(n) for n = 0..10000
J.-P. Allouche, Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence, arXiv:1401.3727 [math.NT], 2014.
J.-P. Allouche, Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence, J. de Théorie des Nombres de Bordeaux, 27, no. 2 (2015), 375-388.
Vladimir Shevelev, Equations of the form t(x+a)=t(x) and t(x+a)=1-t(x) for Thue-Morse sequence, arXiv:0907.0880[math.NT], 2009-2012.

Cf. A161824, A161817, A161674, A161673, A161639, A161641, A161627, A161579, A161580, A121539, A131323, A036554, A010060, A079523, A081706.

tm[0] = 0; tm[n_?EvenQ] := tm[n] = tm[n/2]; tm[n_] := tm[n] = 1 - tm[(n - 1)/2]; Reap[For[n = 0, n <= 18000, n++, If[tm[n] == tm[n + 9], Sow[n]]]][[2, 1]] (* G. C. Greubel, Jan 05 2018 *)
SequencePosition[ThueMorse[Range[0,150]],{x_,,,_,,,_,,,x_}][[All,1]]-1 (* Harvey P. Dale, Feb 06 2023 *)

is(n)=hammingweight(n)%2==hammingweight(n+9)%2 \\ Charles R Greathouse IV, Aug 20 2013

Terms a(35) onward added by G. C. Greubel, Jan 05 2018

A134717 Odd Motzkin numbers.

Original entry on oeis.org

1, 1, 9, 21, 51, 127, 323, 835, 15511, 41835, 853467, 2356779, 50852019, 142547559, 400763223, 1129760415, 3192727797, 9043402501, 208023278209, 593742784829, 1697385471211, 4859761676391, 13933569346707, 40002464776083, 953467954114363, 2750016719520991, 7939655757745265

Offset: 1

Omar E. Pol, Nov 11 2007

A001006 except A134718. - Vladimir Reshetnikov, Nov 02 2015

The asymptotic density of this sequence within the Motzkin numbers is 2/3. - Amiram Eldar, Aug 26 2024

Robert Israel, Table of n, a(n) for n = 1..801

Cf. A001006, A134718, A161674.

S:= series(exp(x)*BesselI(1, 2*x)/x, x, 500):
select(type, [seq(simplify(coeff(S,x,j)*j!), j=0..498)], odd); # Robert Israel, Nov 03 2015

Select[Table[(-1)^n Hypergeometric2F1[3/2, -n, 3, 4], {n, 0, 40}], OddQ] (* Vladimir Reshetnikov, Nov 02 2015 *)

a001006(n) = polcoeff((1-x-sqrt((1-x)^2-4*x^2+x^3*O(x^n)))/ (2*x^2), n); for(n=0, 100, if((m=a001006(n))%2==1, print1(m", "))) \\ Altug Alkan, Nov 03 2015

a(n) = A001006(A161674(n)). - Amiram Eldar, Aug 26 2024

A360949 G.f. A(x) satisfies: 1 = Sum_{n>=0} (-x/2)^n * (A(x)^n + (-1)^n)^n.

Original entry on oeis.org

1, 2, 8, 50, 376, 3124, 27804, 260496, 2539616, 25556330, 263922884, 2785341186, 29948035032, 327315887046, 3630399545244, 40813503158790, 464662514679168, 5354222585965310, 62419468527625408, 736098528973804246, 8781173950238637928, 105987886325647341056

Offset: 0

Paul D. Hanna, Mar 03 2023

nonn

Conjecture: a(0) = 1, a(2*A161674(k) + 1) == 2 (mod 4) for k >= 1, otherwise a(n) == 0 (mod 4). A161674 lists positions n such that A010060(n) + A010060(n+2) = 1, where A010060 is the Thue-Morse sequence.

			G.f.: A(x) = 1 + 2*x + 8*x^2 + 50*x^3 + 376*x^4 + 3124*x^5 + 27804*x^6 + 260496*x^7 + 2539616*x^8 + 25556330*x^9 + ...
such that
1 = 1 - (x/2)*(A(x) - 1) + (x/2)^2*(A(x)^2 + 1)^2 - (x/2)^3*(A(x)^3 - 1)^3 + (x/2)^4*(A(x)^4 + 1)^4 - (x/2)^5*(A(x)^5 - 1)^5 + (x/2)^6*(A(x)^6 + 1)^6 - (x/2)^7*(A(x)^7 - 1)^7 + ...
also,
1 = 2/(2 - x) - 2*x*A(x)/(2 + x*A(x))^2 + 2*x^2*A(x)^4/(2 - x*A(x)^2)^3 - 2*x^3*A(x)^9/(2 + x*A(x)^3)^4 + 2*x^4*A(x)^16/(2 - x*A(x)^4)^5 - 2*x^5*A(x)^25/(2 + x*A(x)^5)^6 + 2*x^6*A(x)^36/(2 - x*A(x)^6)^7 ... + 2*(-x)^n*A(x)^(n^2)/(2 - (-1)^n*x*A(x)^n)^(n+1) + ...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A325574, A161674.

{a(n) = my(A=[1]);
for(i=1, n, A=concat(A, 0); A[#A] = 2*polcoeff( sum(m=0, #A, (-x/2)^m * (Ser(A)^m + (-1)^m)^m ), #A)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) 1 = Sum_{n>=0} (-x/2)^n * (A(x)^n + (-1)^n)^n.
(2) 1 = Sum_{n>=0} 2 * (-x)^n * A(x)^(n^2) / (2 - (-1)^n * x * A(x)^n)^(n+1).
a(n) = A325574(n)/2^n for n >= 0.

A161974 a(n) = number of equalities of the form A010060(n+k) = A010060(n), k=1,2,3.

Original entry on oeis.org

1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1

Offset: 0

Vladimir Shevelev, Jun 23 2009

See comment to A161916. 3-a(n) is the number of equalities of kind A010060(n+k) = 1-A010060(n), k=1,2,3.

Cf. A161916, A161890, A161824, A161817, A161674, A161673, A161639, A161641, A161627, A161579, A161580, A121539, A131323, A036554, A010060.

a(n)=sum(k=1,3,hammingweight(n)%2==hammingweight(n+k)%2) \\ Charles R Greathouse IV, Aug 20 2013

Missing a(24)=1 inserted by Georg Fischer, Jun 21 2024

A162311 Numbers such that A010060(n) = A010060(n+7).

Original entry on oeis.org

1, 3, 4, 5, 7, 10, 14, 17, 19, 20, 21, 23, 25, 27, 28, 29, 31, 33, 35, 36, 37, 39, 42, 46, 49, 51, 52, 53, 55, 58, 62

Offset: 1

Vladimir Shevelev, Jul 01 2009

Or union of intersection of A161673 and {A121539(n)-7} and intersection of A161639 and {A079523(n)-7}.

Conjecture: In every sequence of numbers n, such that A010060(n)=A010060(n+k), for fixed odd k, the odious (A000069) and evil (A001969) terms alternate. - Vladimir Shevelev, Jul 31 2009

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
J.-P. Allouche, Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence, arXiv:1401.3727 [math.NT], 2014.
J.-P. Allouche, Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence, J. de Théorie des Nombres de Bordeaux, 27, no. 2 (2015), 375-388.
V. Shevelev, Equations of the form t(x+a)=t(x) and t(x+a)=1-t(x) for Thue-Morse sequence, arXiv:0907.0880 [math.NT], 2009-2012.

Cf. A161890, A161824, A161817, A161674, A161673, A161639, A161641, A161627, A161579, A161580, A121539, A131323, A036554, A010060, A079523, A081706, A161916, A161974.

tm[0] = 0; tm[n_?EvenQ] := tm[n] = tm[n/2]; tm[n_] := tm[n] = 1 - tm[(n - 1)/2]; Reap[For[n = 0, n <= 20000, n++, If[tm[n] == tm[n + 7], Sow[n]]]][[2, 1]] (* G. C. Greubel, Jan 05 2018 *)

is(n)=hammingweight(n)%2==hammingweight(n+7)%2 \\ Charles R Greathouse IV, Aug 20 2013

A162648 Locations of patterns 1001 or 0110 in the Thue-Morse sequence A010060.

Original entry on oeis.org

0, 4, 6, 8, 12, 16, 20, 22, 24, 28, 30, 32, 36, 38, 40, 44, 48, 52, 54, 56, 60, 64, 68, 70, 72, 76, 80, 84, 86, 88, 92, 94, 96, 100, 102, 104, 108, 112, 116, 118, 120, 124, 126, 128, 132, 134, 136, 140, 144, 148, 150, 152, 156, 158, 160, 164, 166, 168, 172, 176, 180

Offset: 1

Vladimir Shevelev, Jul 08 2009

nonn

Numbers n for which A010060(n+1) = A010060(n+2) = 1-A010060(n) and A010060(n+3) = A010060(n).

Or intersection of A121539, A161674, and A161579.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A161824, A161817, A161674, A161673, A161639, A161641, A161627, A161579, A161580, A121539, A131323, A036554, A010060, A079523, A081706.

Select[Range[3500], MatchQ[IntegerDigits[#, 2], {b : (1) ..} | {_, 0, b : (1) ..} /; OddQ[Length[{b}]]] &] - 1 (* G. C. Greubel, Jan 05 2018 *)
With[{nn=200},Sort[Join[SequencePosition[ThueMorse[Range[0,nn]],{1,0,0,1}],SequencePosition[ ThueMorse[Range[0,nn]],{0,1,1,0}]]][[;;,1]]]-1 (* Harvey P. Dale, Aug 20 2024 *)

is(n)=my(v=vector(4,i,hammingweight(n+i-1))); v[1]==v[4] && v[1]!=v[2] && v[1]!=v[3] \\ Charles R Greathouse IV, Aug 20 2013

a(n) = A079523(n) - 1.

More readable definition from R. J. Mathar, Sep 16 2009

cp's OEIS Frontend

A161817 Positions n such that A010060(n) = A010060(n+5).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A161824 Numbers such that A010060(n) = A010060(n+6).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A161890 Numbers such that A010060(n) = A010060(n+9).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A134717 Odd Motzkin numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A360949 G.f. A(x) satisfies: 1 = Sum_{n>=0} (-x/2)^n * (A(x)^n + (-1)^n)^n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A161974 a(n) = number of equalities of the form A010060(n+k) = A010060(n), k=1,2,3.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Extensions

A162311 Numbers such that A010060(n) = A010060(n+7).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A162648 Locations of patterns 1001 or 0110 in the Thue-Morse sequence A010060.

Views

Author

Keywords

Comments