A026430 -id:A026430 - cp's OEIS frontend

A360397 Intersection of A356133 and A360393.

2, 4, 13, 22, 34, 40, 49, 58, 64, 76, 85, 94, 106, 112, 124, 133, 142, 148, 157, 166, 178, 184, 193, 202, 208, 220, 229, 238, 244, 253, 262, 274, 280, 292, 301, 310, 322, 328, 337, 346, 352, 364, 373, 382, 394, 400, 412, 421, 430, 436, 445, 454, 466, 472

Offset: 1

Clark Kimberling, Feb 10 2023

This is the fourth of four sequences that partition the positive integers. Starting with a general overview, suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers. Let u' and v' be their complements, and assume that the following four sequences are infinite:
(1)  u ^ v = intersection of u and v (in increasing order);
(2)  u ^ v';
(3)  u' ^ v;
(4)  u' ^ v'.
Every positive integer is in exactly one of the four sequences. The limiting densities of these four sequences are 4/9, 2/9, 2/9, and 1/9, respectively.
For A360397, u, v, u', v', are sequences obtained from the Thue-Morse sequence, A026430, as follows:
u = A026530 = (1,3,5,6,8,9,10, 12, ... ) = partial sums of A026430;
u' = A356133 = (2,4,7,11,13,17, 20, ... ) = complement of u;
v = u + 1 = A285954, except its initial 1;
v' = complement of v.

			(1)  u ^ v = (3, 5, 8, 10, 12, 14, 16, 18, 21, 23, 26, 28, 30, 33, ...) =  A360394
(2)  u ^ v' = (1, 6, 9, 15, 19, 24, 27, 31, 36, 42, 45, 51, 55, 60, ...) =  A360395
(3)  u' ^ v = (7, 11, 17, 20, 25, 29, 32, 38, 43, 47, 53, 56, 62, ...) = A360396
(4)  u' ^ v' = (2, 4, 13, 22, 34, 40, 49, 58, 64, 76, 85, 94, 106, ...) = A360397

Cf. A026430, A356133, A360392-A360396, A360398-A360405.

z = 400;
u = Accumulate[1 + ThueMorse /@ Range[0, z]];   (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 2 ; (* A360392 *)
v1 = Complement[Range[Max[v]], v];    (* A360393 *)
Intersection[u, v]    (* A360394 *)
Intersection[u, v1]   (* A360395 *)
Intersection[u1, v]   (* A360396 *)
Intersection[u1, v1]  (* A360397 *)

A360401 a(n) = A356133(A360393(n)).

Original entry on oeis.org

2, 4, 11, 17, 25, 38, 43, 56, 64, 71, 79, 92, 101, 106, 119, 124, 133, 146, 151, 164, 173, 178, 191, 197, 206, 218, 227, 233, 242, 253, 260, 272, 280, 287, 295, 308, 317, 322, 335, 341, 350, 362, 371, 377, 385, 398, 403, 415, 425, 430, 443, 449, 457, 470

Offset: 1

Clark Kimberling, Mar 11 2023

This is the third of four sequences that partition the positive integers. Suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers.   Let u' and v' be their (increasing) complements, and consider these four sequences:
(1) u o v, defined by (u o v)(n)  = u(v(n));
(2) u o v';
(3) u' o v;
(4) v' o u'.
Every positive integer is in exactly one of the four sequences. Their limiting densities are 4/9, 2/9, 2/9, 1/9 (and likewise for A360394-A360397 and A360402-A360405).

			(1)  u o v = (5, 8, 10, 12, 15, 16, 18, 21, 24, 26, 27, 30, 31, 35, 37, 39, ...) = A360398
(2)  u o v' = (1, 3, 6, 9, 14, 19, 23, 28, 33, 36, 41, 46, 51, 54, 60, 63, 68, ...) = A360399
(3)  u' o v = (7, 13, 20, 22, 29, 32, 34, 40, 47, 49, 53, 58, 62, 67, 74, 76, ...) = A360400
(4)  u' o v' = (2, 4, 11, 17, 25, 38, 43, 56, 64, 71, 79, 92, 101, 106, 119, ...) = A360401

Cf. A026530, A356133, A360392, A360393, A360398, A286354, A286355, A360394 (intersections instead of results of composition), A360402-A360405.

z = 2000;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 2;  (* A360392 *)
v1 = Complement[Range[Max[v]], v];  (* A360393 *)
zz = 100;
Table[u[[v[[n]]]], {n, 1, zz}]    (* A360398 *)
Table[u[[v1[[n]]]], {n, 1, zz}]   (* A360399 *)
Table[u1[[v[[n]]]], {n, 1, zz}]   (* A360400 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]  (* A360401 *)

A076826 a(n) = 2*(Sum_{k=0..n} A010060(k)) - n, where A010060 is a Thue-Morse sequence.

Original entry on oeis.org

0, 1, 2, 1, 2, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 1, 2, 1, 2, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 1, 2, 1, 2, 1, 0, 1, 0, 1, 2, 1, 2, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 1, 2, 1, 2, 1, 0, 1, 0, 1, 2, 1, 2, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 1, 2, 1, 2, 1, 0, 1, 2, 1

Offset: 0

Benoit Cloitre, Nov 24 2002

nonn

Is there any interesting sequence b(n) such that b(n) mod 3 = a(n)?

Fixed point of the morphism 0->012; 1->1; 2->210 starting with a(0) = 0. - Philippe Deléham, Mar 14 2004

Antti Karttunen, Table of n, a(n) for n = 0..8191
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 52.
Index entries for sequences that are fixed points of mappings

Cf. A000069 (odious numbers), A001969 (evil numbers).

Cf. A000120, A007413, A010060, A026430, A115384, A159481.

Nest[ Function[ l, {Flatten[(l /. {0 -> {0, 1, 2}, 1 -> {1}, 2 -> {2, 1, 0}}) ]}], {0}, 6] (* Robert G. Wilson v, Mar 03 2005 *)
cnt=0; Join[{0}, Table[If[EvenQ[Count[IntegerDigits[n,2],1]], cnt--, cnt++ ]; cnt, {n,150}]] (* T. D. Noe, Jun 14 2007 *)

a(n)=if(n<0,0,2*sum(k=1,n,subst(Pol(binary(k)),x,1)%2)-n)

a(n)=if(n<1,0,if(n%2,1,if(n/2%2,2-a(n\4*2),a(n/2))))

def A076826(n): return 1 if n&1 else (n.bit_count()&1)<<1 # Chai Wah Wu, Mar 01 2023

a(2k+1) = 1, a(4k) = a(2k), a(4k+2) = 2-a(2k). - Michael Somos, Dec 04 2002
a(2n) = 2*A010060(n); a(2n+1) = 1. - Benoit Cloitre, Mar 08 2004
a(n) = 2*(A026430(n+1) - 1) mod 3. - Philippe Deléham, Mar 28 2004
a(n) = (number of odious numbers <= n) - (number of evil numbers <= n) for n>0. - T. D. Noe, Jun 14 2007
a(n) = 2*A115384(n) - n. - Vladimir Shevelev, May 31 2009
a(n) = 0 if n and A000120(n) are even; a(n) = 2 if n is even but A000120(n) is odd; a(n) = 1 if n is odd. - Vladimir Shevelev, May 31 2009

A360135 a(n) = A356133(A285953(n+1)).

Original entry on oeis.org

2, 7, 13, 22, 34, 40, 53, 62, 67, 76, 89, 97, 104, 115, 122, 131, 142, 148, 161, 169, 176, 187, 193, 202, 215, 223, 229, 238, 251, 257, 269, 278, 283, 292, 305, 313, 320, 331, 337, 346, 359, 367, 373, 382, 394, 400, 412, 421, 428, 439, 445, 454, 466, 472

Offset: 1

Clark Kimberling, Jan 30 2023

This is the fourth of four sequences that partition the positive integers. Suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers. Let u' and v' be their (increasing) complements, and consider these four sequences:
(1) u o v, defined by (u o v)(n) = u(v(n));
(2) u o v';
(3) u' o v;
(4) v' o u'.
Every positive integer is in exactly one of the four sequences. Their limiting densities are 4/9, 2/9, 2/9, 1/9, respectively.

			(1) u o v = (3, 6, 9, 10, 14, 15, 16, 19, 23, 24, 26, 28, 30, 33, 36, 37, 41, ...) = A359352
(2) u o v' = (1, 5, 8, 12, 18, 21, 27, 31, 35, 39, 45, 50, 52, 59, 61, 66, 72, ...) = A359353
(3) u' o v = (4, 11, 17, 20, 25, 29, 32, 38, 43, 47, 49, 56, 58, 64, 71, 74, ...) = A360134
(4) u' o v' = (2, 7, 13, 22, 34, 40, 53, 62, 67, 76, 89, 97, 104, 115, 122, ...) = A360135

Cf. A026530, A359352, A285953, A285954, A359277 (intersections instead of results of composition), A359352-A360134, A360136-A360139.

z = 2000; zz = 100;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 1;  (* A285954 *)
v1 = Complement[Range[Max[v]], v];  (* A285953 *)
Table[u[[v[[n]]]], {n, 1, zz}]      (* A359352 *)
Table[u[[v1[[n]]]], {n, 1, zz}]     (* A359353 *)
Table[u1[[v[[n]]]], {n, 1, zz}]     (* A360134 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]    (* A360135 *)

A360396 Intersection of A356133 and A360392.

Original entry on oeis.org

7, 11, 17, 20, 25, 29, 32, 38, 43, 47, 53, 56, 62, 67, 71, 74, 79, 83, 89, 92, 97, 101, 104, 110, 115, 119, 122, 127, 131, 137, 140, 146, 151, 155, 161, 164, 169, 173, 176, 182, 187, 191, 197, 200, 206, 211, 215, 218, 223, 227, 233, 236, 242, 247, 251, 257

Offset: 1

Clark Kimberling, Feb 05 2023

This is the third of four sequences that partition the positive integers. Starting with a general overview, suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers.  Let u' and v' be their complements, and assume that the following four sequences are infinite:
(1)  u ^ v = intersection of u and v (in increasing order);
(2)  u ^ v';
(3)  u' ^ v;
(4)  u' ^ v'.
Every positive integer is in exactly one of the four sequences. The limiting densities of these four sequences are 4/9, 2/9, 2/9, and 1/9, respectively.
For A360396, u, v, u', v', are sequences obtained from the Thue-Morse sequence, A026430, as follows:
u = A026530 = (1,3,5,6,8,9,10, 12, ... ) = partial sums of A026430
u' = A356133 = (2,4,7,11,13,17, 20, ... ) = complement of u
v = u + 1 = A285954, except its initial 1
v' = complement of v.

			(1)  u ^ v = (3, 5, 8, 10, 12, 14, 16, 18, 21, 23, 26, 28, 30, 33, ...) =  A360394
(2)  u ^ v' = (1, 6, 9, 15, 19, 24, 27, 31, 36, 42, 45, 51, 55, 60, ...) =  A360395
(3)  u' ^ v = (7, 11, 17, 20, 25, 29, 32, 38, 43, 47, 53, 56, 62, ...) = A360396
(4)  u' ^ v' = (2, 4, 13, 22, 34, 40, 49, 58, 64, 76, 85, 94, 106, ...) = A360397

Cf. A026430, A356133, A360392-A360395, A360397-A360405.

z = 400;
u = Accumulate[1 + ThueMorse /@ Range[0, z]];   (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 2 ; (* A360392 *)
v1 = Complement[Range[Max[v]], v];  (* A360393 *)
Intersection[u, v]    (* A360394 *)
Intersection[u, v1]   (* A360395 *)
Intersection[u1, v]   (* A360396 *)
Intersection[u1, v1]  (* A360397 *)

A360400 a(n) = A356133(A360392(n)).

Original entry on oeis.org

7, 13, 20, 22, 29, 32, 34, 40, 47, 49, 53, 58, 62, 67, 74, 76, 83, 85, 89, 94, 97, 104, 110, 112, 115, 122, 127, 131, 137, 140, 142, 148, 155, 157, 161, 166, 169, 176, 182, 184, 187, 193, 200, 202, 208, 211, 215, 220, 223, 229, 236, 238, 244, 247, 251, 257

Offset: 1

Clark Kimberling, Mar 11 2023

This is the third of four sequences that partition the positive integers. Suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers.   Let u' and v' be their (increasing) complements, and consider these four sequences:
(1) u o v, defined by (u o v)(n)  = u(v(n));
(2) u o v';
(3) u' o v;
(4) v' o u'.
Every positive integer is in exactly one of the four sequences. Their limiting densities are 4/9, 2/9, 2/9, 1/9 (and likewise for A360394-A360397 and A360402-A360405).

			(1)  u o v = (5, 8, 10, 12, 15, 16, 18, 21, 24, 26, 27, 30, 31, 35, 37, 39, ...) = A360398
(2)  u o v' = (1, 3, 6, 9, 14, 19, 23, 28, 33, 36, 41, 46, 51, 54, 60, 63, 68, ...) = A360399
(3)  u' o v = (7, 13, 20, 22, 29, 32, 34, 40, 47, 49, 53, 58, 62, 67, 74, 76, ...) = A360400
(4)  u' o v' = (2, 4, 11, 17, 25, 38, 43, 56, 64, 71, 79, 92, 101, 106, 119, ...) = A360401

Cf. A026530, A356133, A360392, A360393, A360398, A286354, A286356, A360394 (intersections instead of results of composition), A360402-A360405.

z = 2000;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 2;  (* A360392 *)
v1 = Complement[Range[Max[v]], v];  (* A360393 *)
zz = 100;
Table[u[[v[[n]]]], {n, 1, zz}]    (* A360398 *)
Table[u[[v1[[n]]]], {n, 1, zz}]   (* A360399 *)
Table[u1[[v[[n]]]], {n, 1, zz}]   (* A360400 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]  (* A360401 *)

A360404 a(n) = A360392(A356133(n)).

Original entry on oeis.org

5, 8, 12, 18, 21, 28, 32, 35, 39, 46, 50, 53, 59, 62, 67, 72, 75, 82, 86, 89, 95, 98, 102, 109, 113, 116, 120, 127, 130, 136, 140, 143, 147, 154, 158, 161, 167, 170, 174, 181, 185, 188, 192, 198, 201, 207, 212, 215, 221, 224, 228, 234, 237, 243, 248, 251

Offset: 1

Clark Kimberling, Apr 01 2023

This is the third of four sequences that partition the positive integers. Suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers. Let u' and v' be their (increasing) complements, and consider these four sequences:
(1) v o u, defined by (v o u)(n) = v(u(n));
(2) v' o u;
(3) v o u';
(4) v' o u.
Every positive integer is in exactly one of the four sequences. Their limiting densities are 4/9, 2/9, 2/9, 1/9, respectively (and likewise for A360394-A360401).

			(1)  v o u = (3, 7, 10, 11, 14, 16, 17, 20, 23, 25, 26, 29, 30, 33, 37, ...) = A360402
(2)  v' o u = (1, 4, 9, 13, 19, 22, 24, 31, 36, 40, 42, 49, 51, 58, 64, ...) = A360403
(3)  v o u' = (5, 8, 12, 18, 21, 28, 32, 35, 39, 46, 50, 53, 59, 62, 67, ...) = A360404
(4)  v' o u' = (2, 6, 15, 27, 34, 45, 55, 60, 69, 81, 91, 96, 108, 114, ...) = A360405

Cf. A026530, A360392, A360393, A360394-A3546352 (intersections instead of results of compositions), A360398-A360401 (results of reversed compositions), A360402, A360403, A360405.

z = 2000; zz = 100;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 2;  (* A360392 *)
v1 = Complement[Range[Max[v]], v];  (* A360393 *)
Table[v[[u[[n]]]], {n, 1, zz}]     (* A360402 *)
Table[v1[[u[[n]]]], {n, 1, zz}]    (* A360403 *)
Table[v[[u1[[n]]]], {n, 1, zz}]    (* A360404 *)
Table[v1[[u1[[n]]]], {n, 1, zz}]   (* A360405 *)

A354384 Difference sequence of A356133.

Original entry on oeis.org

2, 3, 4, 2, 4, 3, 2, 3, 4, 3, 2, 4, 2, 3, 4, 2, 4, 3, 2, 4, 2, 3, 4, 3, 2, 3, 4, 2, 4, 3, 2, 3, 4, 3, 2, 4, 2, 3, 4, 3, 2, 3, 4, 2, 4, 3, 2, 4, 2, 3, 4, 2, 4, 3, 2, 3, 4, 3, 2, 4, 2, 3, 4, 2, 4, 3, 2, 4, 2, 3, 4, 3, 2, 3, 4, 2, 4, 3, 2, 4, 2, 3, 4, 2, 4, 3

Offset: 1

Clark Kimberling, Aug 04 2022

Cf. A026430, A356133, A091855 (positions of 2), A036554 (positions of 3), A091855 (positions of 4).

u = Accumulate[1 + ThueMorse /@ Range[0, 200]]  (* A026430 *)
v = Complement[Range[Max[u]], u];  (* A356133 *)
Differences[v] (* A354384 *)

a(n) = A007413(n) + 1.

a(n) = A036580(n) + 2.

A088564 a(n)=sum(i=0,n,binomial(2*i,i) (mod 3)).

Original entry on oeis.org

1, 3, 3, 5, 6, 6, 6, 6, 6, 8, 9, 9, 10, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 14, 15, 15, 16, 18, 18, 18, 18, 18, 19, 21, 21, 23, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24

Offset: 0

Benoit Cloitre, Nov 19 2003

nonn

Distinct values (i.e. 1,3,5,6,8,9,...) are given by the partial sums of the Thue-Morse sequence on alphabet (1,2) A026430. Sequence of least k such that a(k)>a(k-1) is given by A005836. For any k>=0, card{ n : a(3*A005836(k)) =a(n)}=1.

Only 79 of the first 1001 terms are odd numbers. -- From Harvey P. Dale, Aug 08 2012

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A001285, A005836, A006996.

Table[Sum[Mod[Binomial[2*i,i],3],{i,0,n}],{n,0,80}] (* Harvey P. Dale, Aug 08 2012 *)

cp's OEIS Frontend

A360397 Intersection of A356133 and A360393.

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A360401 a(n) = A356133(A360393(n)).

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A076826 a(n) = 2*(Sum_{k=0..n} A010060(k)) - n, where A010060 is a Thue-Morse sequence.

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A360135 a(n) = A356133(A285953(n+1)).

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A360396 Intersection of A356133 and A360392.

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A360400 a(n) = A356133(A360392(n)).

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A360404 a(n) = A360392(A356133(n)).

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A354384 Difference sequence of A356133.

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A088564 a(n)=sum(i=0,n,binomial(2*i,i) (mod 3)).

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