A287555 -id:A287555 - cp's OEIS frontend

A053839 a(n) = (sum of digits of n written in base 4) modulo 4.

0, 1, 2, 3, 1, 2, 3, 0, 2, 3, 0, 1, 3, 0, 1, 2, 1, 2, 3, 0, 2, 3, 0, 1, 3, 0, 1, 2, 0, 1, 2, 3, 2, 3, 0, 1, 3, 0, 1, 2, 0, 1, 2, 3, 1, 2, 3, 0, 3, 0, 1, 2, 0, 1, 2, 3, 1, 2, 3, 0, 2, 3, 0, 1, 1, 2, 3, 0, 2, 3, 0, 1, 3, 0, 1, 2, 0, 1, 2, 3, 2, 3, 0, 1, 3, 0, 1, 2, 0, 1, 2, 3, 1, 2, 3, 0, 3, 0, 1, 2, 0, 1, 2, 3, 1

Offset: 0

Henry Bottomley, Mar 28 2000

a(n) is the third row of the array in A141803. - Andrey Zabolotskiy, May 16 2016

This is the fixed point of the morphism 0->0123, 1->1230, 2->2301, 3->3012 starting with 0. Let t be the (nonperiodic) sequence of positions of 0, and likewise, u for 1, v for 2, and w for 3; then t(n)/n -> 4, u(n)/n -> 4, v(n)/n -> 4, w(n)/n -> 4, and t(n) + u(n) + v(n) + w(n) = 16*n - 6 for n >= 1. - Clark Kimberling, May 31 2017

			First three iterations of the morphism 0->0123, 1->1230, 2->2301, 3->3012:
  0123
  0123123023013012
  0123123023013012123023013012012323013012012312303012012312302301

Robert Israel, Table of n, a(n) for n = 0..10000
Glen Joyce C. Dulatre, Jamilah V. Alarcon, Vhenedict M. Florida and Daisy Ann A. Disu, On Fractal Sequences, DMMMSU-CAS Science Monitor (2016-2017) Vol. 15 No. 2, 109-113.
Robert Walker, Self Similar Sloth Canon Number Sequences
Index entries for sequences that are fixed points of mappings

Cf. A010060, A053837-A053844, A141803, A287552, A287553, A287554, A287555.

Cf. A010873, A053737.

seq(convert(convert(n,base,4),`+`) mod 4, n=0..100); # Robert Israel, May 18 2016

Mod[Total@ IntegerDigits[#, 4], 4] & /@ Range[0, 120] (* Michael De Vlieger, May 17 2016 *)
s = Nest[Flatten[# /. {0 -> {0, 1, 2, 3}, 1 -> {1, 2, 3, 0}, 2 -> {2, 3, 0, 1}, 3 -> {3, 0, 1, 2}}] &, {0}, 9];   (* - Clark Kimberling, May 31 2017 *)

a(n) = vecsum(digits(n,4)) % 4; \\ Michel Marcus, May 16 2016

a(n) = sumdigits(n, 4) % 4; \\ Michel Marcus, Jul 04 2018

a(n) = A010873(A053737(n)). - Andrey Zabolotskiy, May 18 2016

G.f. G(x) satisfies x^81*G(x) - (x^72+x^75+x^78+x^81)*G(x^4) + (x^48+x^60+x^63-x^64+x^72+x^75-x^76+x^78-x^79-x^88-x^91-x^94)*G(x^16) + (-1+x^16-x^48-x^60-x^63+2*x^64+x^76+x^79-x^80+x^112+x^124+x^127-x^128-x^140-x^143)*G(x^64) + (1-x^16-x^64+x^80-x^256+x^272+x^320-x^336)*G(x^256) = 0. - Robert Israel, May 18 2016

A287556 Start with 0 and repeatedly substitute 0->0132, 1->1320, 2->3201, 3->2013.

Original entry on oeis.org

0, 1, 3, 2, 1, 3, 2, 0, 2, 0, 1, 3, 3, 2, 0, 1, 1, 3, 2, 0, 2, 0, 1, 3, 3, 2, 0, 1, 0, 1, 3, 2, 3, 2, 0, 1, 0, 1, 3, 2, 1, 3, 2, 0, 2, 0, 1, 3, 2, 0, 1, 3, 3, 2, 0, 1, 0, 1, 3, 2, 1, 3, 2, 0, 1, 3, 2, 0, 2, 0, 1, 3, 3, 2, 0, 1, 0, 1, 3, 2, 3, 2, 0, 1, 0, 1

Offset: 1

Clark Kimberling, May 31 2017

This is the fixed point of the morphism 0->0132, 1->1320, 2->3201, 3->2013 starting with 0.  Let t be the (nonperiodic) sequence of positions of 0, and likewise, u for 1, v for 2, and w for 3; then t(n)/n -> 4, u(n)/n -> 4, v(n)/n -> 4,  w(n)/n -> 4.   Also, t(n) + u(n) + v(n) + w(n) = 16*n - 6 for n >= 1.
In the following guide to related sequences, column 1 indexes fixed points on {0,1,2,3}, and column 2 indicates position sequences of 0, 1, 2, 3.  Those sequences therefore comprise a 4-way splitting of the positive integers.
Fixed points of morphisms:                        Position sequences:
A053839: 0->0123, 1->1230, 2->2301, 3->3012       A287552-A287555
A287556: 0->0132, 1->1320, 2->3201, 3->2013       A287557-A287560
A287561: 0->0213, 1->2130, 2->1302, 3->3021       A287562-A287565
A287566: 0->0231, 1->2310, 2->3102, 3->1023       A287567-A287570
A287571: 0->0312, 1->3120, 2->1203, 3->2031       A287572-A287575
A287576: 0->0321, 1->3210, 2->2103, 3->1032       A287577-A287580

			First three iterations of the morphism:
0132
0132132020133201
0132132020133201132020133201013232010132132020132013320101321320

Clark Kimberling, Table of n, a(n) for n = 1..20000
Index entries for sequences that are fixed points of mappings

Cf. A287557, A287558, A287559, A287560.

s = Nest[Flatten[# /. {0 -> {0, 1, 3, 2}, 1 -> {1, 3, 2, 0}, 2 -> {3, 2, 0, 1}, 3 -> {2, 0, 1, 3}}] &, {0}, 9]; (* A287556 *)
Flatten[Position[s, 0]]; (* A287557 *)
Flatten[Position[s, 1]]; (* A287558 *)
Flatten[Position[s, 2]]; (* A287559 *)
Flatten[Position[s, 3]]; (* A287560 *)

A287552 Positions of 0 in A053839.

Original entry on oeis.org

1, 8, 11, 14, 20, 23, 26, 29, 35, 38, 41, 48, 50, 53, 60, 63, 68, 71, 74, 77, 83, 86, 89, 96, 98, 101, 108, 111, 113, 120, 123, 126, 131, 134, 137, 144, 146, 149, 156, 159, 161, 168, 171, 174, 180, 183, 186, 189, 194, 197, 204, 207, 209, 216, 219, 222, 228

Offset: 1

Clark Kimberling, May 31 2017

a(n) - a(n-1) is in {1,2,3,4,5,6,7} for n >= 1; also, 4n - a(n) is in {0,1,2,3} for n >= 1.  The first 20 numbers 4n - a(n) are 3, 0, 1, 2, 0, 1, 2, 3, 1, 2, 3, 0, 2, 3, 0, 1, 0, 1, 2, 3, with
in positions given by A287553,
in positions given by A287554,
in positions given by A287555,
in positions given by A287552.

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

Cf. A053839, A287553, A287554, A287555.

s = Nest[Flatten[# /. {0 -> {0, 1, 2, 3}, 1 -> {1, 2, 3, 0}, 2 -> {2, 3, 0, 1}, 3 -> {3, 0, 1, 2}}] &, {0}, 9];  (* A053839 *)
Flatten[Position[s, 0]]; (* A287552 *)
Flatten[Position[s, 1]]; (* A287553 *)
Flatten[Position[s, 2]]; (* A287554 *)
Flatten[Position[s, 3]]; (* A287555 *)

A287553 Positions of 1 in A053839.

Original entry on oeis.org

2, 5, 12, 15, 17, 24, 27, 30, 36, 39, 42, 45, 51, 54, 57, 64, 65, 72, 75, 78, 84, 87, 90, 93, 99, 102, 105, 112, 114, 117, 124, 127, 132, 135, 138, 141, 147, 150, 153, 160, 162, 165, 172, 175, 177, 184, 187, 190, 195, 198, 201, 208, 210, 213, 220, 223, 225

Offset: 1

Clark Kimberling, May 31 2017

a(n) - a(n-1) is in {1,2,3,4,5,6,7} for n >= 1; also, 4n - a(n) is in {0,1,2,3} for n >= 1.  The first 20 numbers 4n - a(n) are 2, 3, 0, 1, 3, 0, 1, 2, 0, 1, 2, 3, 1, 2, 3, 0, 3, 0, 1, 2, with
in positions given by A287554,
in positions given by A287555,
in positions given by A287552,
in positions given by A287553.

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

Cf. A053839, A287552, A287554, A287555.

s = Nest[Flatten[# /. {0 -> {0, 1, 2, 3}, 1 -> {1, 2, 3, 0}, 2 -> {2, 3, 0, 1}, 3 -> {3, 0, 1, 2}}] &, {0}, 9];         (* A053839 *)
Flatten[Position[s, 0]]; (* A287552 *)
Flatten[Position[s, 1]]; (* A287553 *)
Flatten[Position[s, 2]]; (* A287554 *)
Flatten[Position[s, 3]]; (* A287555 *)

A287554 Positions of 2 in A053839.

Original entry on oeis.org

3, 6, 9, 16, 18, 21, 28, 31, 33, 40, 43, 46, 52, 55, 58, 61, 66, 69, 76, 79, 81, 88, 91, 94, 100, 103, 106, 109, 115, 118, 121, 128, 129, 136, 139, 142, 148, 151, 154, 157, 163, 166, 169, 176, 178, 181, 188, 191, 196, 199, 202, 205, 211, 214, 217, 224, 226

Offset: 1

Clark Kimberling, May 31 2017

a(n) - a(n-1) is in {1,2,3,4,5,6,7} for n >= 1; also, 4n - a(n) is in {0,1,2,3} for n >= 1.  The first 20 numbers 4n - a(n) are 1, 2, 3, 0, 2, 3, 0, 1, 3, 0, 1, 2, 0, 1, 2, 3, 2, 3, 0, 1, with
in positions given by A287555,
in positions given by A287552,
in positions given by A287553,
in positions given by A287554.

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

Cf. A053839, A287552, A287553, A287555.

s = Nest[Flatten[# /. {0 -> {0, 1, 2, 3}, 1 -> {1, 2, 3, 0}, 2 -> {2, 3, 0, 1}, 3 -> {3, 0, 1, 2}}] &, {0}, 9];  (* A053839 *)
Flatten[Position[s, 0]]; (* A287552 *)
Flatten[Position[s, 1]]; (* A287553 *)
Flatten[Position[s, 2]]; (* A287554 *)
Flatten[Position[s, 3]]; (* A287555 *)

cp's OEIS Frontend

A053839 a(n) = (sum of digits of n written in base 4) modulo 4.

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A287556 Start with 0 and repeatedly substitute 0->0132, 1->1320, 2->3201, 3->2013.

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A287552 Positions of 0 in A053839.

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A287553 Positions of 1 in A053839.

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A287554 Positions of 2 in A053839.

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