A026601 -id:A026601 - cp's OEIS frontend

A026600 a(n) is the n-th letter of the infinite word generated from w(1)=1 inductively by w(n)=JUXTAPOSITION{w(n-1),w'(n-1),w"(n-1)}, where w(k) becomes w'(k) by the cyclic permutation 1->2->3->1 and w"(k) = (w')'(k).

1, 2, 3, 2, 3, 1, 3, 1, 2, 2, 3, 1, 3, 1, 2, 1, 2, 3, 3, 1, 2, 1, 2, 3, 2, 3, 1, 2, 3, 1, 3, 1, 2, 1, 2, 3, 3, 1, 2, 1, 2, 3, 2, 3, 1, 1, 2, 3, 2, 3, 1, 3, 1, 2, 3, 1, 2, 1, 2, 3, 2, 3, 1, 1, 2, 3, 2, 3, 1, 3, 1, 2, 2, 3, 1, 3, 1, 2, 1, 2, 3, 2, 3, 1, 3, 1, 2, 1, 2, 3, 3, 1, 2, 1, 2, 3, 2, 3, 1, 1, 2, 3, 2, 3, 1

Offset: 1

Clark Kimberling

nonn

			1;
(123);
(123)(231)(312);
(123)(231)(312)(231)(312)(123)(312)(123)(231);

Michael Gilleland, Some Self-Similar Integer Sequences
Index entries for sequences that are fixed points of mappings

Equals A053838(n-1) + 1. Cf. A026601-A026614.

Nest[ Flatten[ # /. {1 -> {1, 2, 3}, 2 -> {2, 3, 1}, 3 -> {3, 1, 2}}] &, {1}, 7] (* Robert G. Wilson v, Mar 08 2005 *)

{a(n) = if( n<2, n>0, (a((n + 2)\ 3) + n + 1 )%3 + 1)} /* Michael Somos, Sep 06 2008 */

a(A026601(n)) = 1.
a(A026602(n)) = 2.
a(A026603(n)) = 3. -Michael Somos, Sep 06 2008

A026602 Numbers k such that A026600(k) = 2.

Original entry on oeis.org

2, 4, 9, 10, 15, 17, 21, 23, 25, 28, 33, 35, 39, 41, 43, 47, 49, 54, 57, 59, 61, 65, 67, 72, 73, 78, 80, 82, 87, 89, 93, 95, 97, 101, 103, 108, 111, 113, 115, 119, 121, 126, 127, 132, 134, 137, 139, 144, 145, 150, 152, 156, 158, 160

Offset: 1

Clark Kimberling

nonn

It appears that a(n) gives the position of its own n-th 1 modulo 3 term, the n-th 2 modulo 3 term in A026603, and the n-th multiple of 3 in A026601. A026601 and A026603 appear to have the same sort of indexical properties. - Matthew Vandermast, Oct 06 2010

Cf. A026600, A026601, A026603.

A026603 Numbers k such that A026600(k) = 3.

Original entry on oeis.org

3, 5, 7, 11, 13, 18, 19, 24, 26, 29, 31, 36, 37, 42, 44, 48, 50, 52, 55, 60, 62, 66, 68, 70, 74, 76, 81, 83, 85, 90, 91, 96, 98, 102, 104, 106, 109, 114, 116, 120, 122, 124, 128, 130, 135, 138, 140, 142, 146, 148, 153, 154, 159, 161

Offset: 1

Clark Kimberling

nonn

It appears that a(n) gives the position of its own n-th 1 modulo 3 term, the n-th 2 modulo 3 term in A026601, and the n-th multiple of 3 in A026602. A026601 and A026602 appear to have the same sort of indexical properties. - Matthew Vandermast, Oct 06 2010

Cf. A026600, A026601, A026602.

A079498 Numbers whose sum of digits in base b gives 0 (mod b), for b = 3.

Original entry on oeis.org

0, 5, 7, 11, 13, 15, 19, 21, 26, 29, 31, 33, 37, 39, 44, 45, 50, 52, 55, 57, 62, 63, 68, 70, 74, 76, 78, 83, 85, 87, 91, 93, 98, 99, 104, 106, 109, 111, 116, 117, 122, 124, 128, 130, 132, 135, 140, 142, 146, 148, 150, 154, 156, 161, 163, 165, 170, 171, 176, 178, 182, 184

Offset: 1

Carlos Alves, Jan 21 2003

In base 2 this gives the "Evil Numbers" (cf. A001969). One may conjecture that in base b the asymptotic slope will be b and asymptotic density 1/b for each result (mod b). Cases b=31 or b=61 gave considerable number of primes on the sequence.
Proof of this conjecture: in general, the sequence d with terms d(n) = sum of digits of n written in base b (mod b) is a fixed point of the generalized Thue-Morse morphism 0->01..b-1, 1->12..0, etc. See A053839 for the case b=4. It follows directly from this that all symbols have asymptotic density 1/b, and therefore that the positional sequences all have asymptotic slope b. - Michel Dekking, Apr 18 2019
Positions of 0's in A053838. Cf. A026601.

			83 is a term since 83 = (1,0,0,0,2)_3 and 1 + 0 + 0 + 0 + 2 = 3 == 0 (mod 3).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001969. See A053840 for base b=5. See A141803 for an array with all b.

Ev = Function[{b, x}, vx = IntegerDigits[x, b]; Mod[Apply[Plus, vx], b]]; Seq = Function[{b, n}, Flatten[Position[Table[Ev[b, k], {k, 0, n}], 0]] - 1]; sb = Seq[3, 1000]

a(1) = 0 inserted and offset corrected by Amiram Eldar, Jan 05 2020

A287435 Positions of 0 in A053838.

Original entry on oeis.org

1, 6, 8, 12, 14, 16, 20, 22, 27, 30, 32, 34, 38, 40, 45, 46, 51, 53, 56, 58, 63, 64, 69, 71, 75, 77, 79, 84, 86, 88, 92, 94, 99, 100, 105, 107, 110, 112, 117, 118, 123, 125, 129, 131, 133, 136, 141, 143, 147, 149, 151, 155, 157, 162, 164, 166, 171, 172, 177

Offset: 1

Clark Kimberling, May 26 2017

a(n) - a(n-1) is in {1,2,3,4,5} for n >= 1; also, 3n - a(n) is in {0, 1, 2} for n >= 1. The first 20 numbers 3n - a(n) are 2, 0, 1, 0, 1, 2, 1, 2, 0, 0, 1, 2, 1, 2, 0, 2, 0, 1, 1, 2, with 0 in positions given by A287436, 1 is in positions given by A287437, and 2 in positions given by A287435.

Does this differ from A026601? - R. J. Mathar, Jun 14 2017

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A053838, A287436, A287437.

s = Nest[Flatten[# /. {0->{0, 1, 2}, 1->{1, 2, 0}, 2->{2, 0, 1}}] &, {0}, 9]; (* A053838 *)
Flatten[Position[s, 0]]; (* A287435 *)
Flatten[Position[s, 1]]; (* A287436 *)
Flatten[Position[s, 2]]; (* A287437 *)

The definition refers to a different offset in A053838. - R. J. Mathar, May 30 2017

cp's OEIS Frontend

A026600 a(n) is the n-th letter of the infinite word generated from w(1)=1 inductively by w(n)=JUXTAPOSITION{w(n-1),w'(n-1),w"(n-1)}, where w(k) becomes w'(k) by the cyclic permutation 1->2->3->1 and w"(k) = (w')'(k).

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A026602 Numbers k such that A026600(k) = 2.

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A026603 Numbers k such that A026600(k) = 3.

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A079498 Numbers whose sum of digits in base b gives 0 (mod b), for b = 3.

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A287435 Positions of 0 in A053838.

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