A026603 -id:A026603 - 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

A026601 Numbers k such that A026600(k) = 1.

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

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 A026602, and the n-th multiple of 3 in A026603. A026602 and A026603 appear to have analogous indexical properties. - Matthew Vandermast, Oct 06 2010

This follows directly from the generating morphism for A026600: a 1 in position k creates a 1 in position 3k-2, a 2 in position 3k-1, and a 3 in position 3k. Since each block of three terms in A026600 is a permutation of {1,2,3}, these created terms are the k-th terms of their respective index sequences. The proof for the other index sequences is similar. - Charlie Neder, Mar 10 2019

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

Cf. A026600, A026602, A026603, A079498.

a(n) = A079498(n) + 1.

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.

A287437 Positions of 2 in A053838.

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, 163, 168, 170, 174, 176

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.

Does this differ from A026603? - R. J. Mathar, Jun 14 2017

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

Cf. A053838, A287435, A287436.

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*)

a(n) + A053838(n-1) = 3n.

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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A026601 Numbers k such that A026600(k) = 1.

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

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A287437 Positions of 2 in A053838.

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