A088950 -id:A088950 - cp's OEIS frontend

A088952 Numbers that are squarefree words in ternary representation.

0, 1, 2, 3, 5, 6, 7, 10, 11, 15, 16, 19, 20, 21, 23, 32, 33, 34, 46, 47, 48, 57, 59, 61, 64, 65, 69, 96, 97, 100, 102, 104, 138, 140, 142, 145, 146, 173, 177, 178, 183, 185, 194, 195, 196, 208, 209, 289, 290, 291, 300, 302, 307, 312, 416, 421, 426, 428, 437, 438

Offset: 1

Reinhard Zumkeller, Oct 25 2003

A088950(a(n)) = 0.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A088952
Eric Weisstein's World of Mathematics, Squarefree Word

Cf. A007089, A088953.

Select[Range[0, 500], MatchQ[IntegerDigits[#, 3], Except[{_, x__, x__, _}]] &] (* Vladimir Reshetnikov, May 17 2016 *)

PARI
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See Links section.
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A333124 a(n) is the number of square-subwords in the binary representation of n.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 1, 2, 2, 1, 1, 1, 2, 1, 2, 4, 4, 2, 1, 2, 2, 2, 1, 2, 3, 2, 2, 2, 3, 2, 4, 6, 6, 4, 2, 3, 3, 2, 2, 3, 3, 2, 3, 3, 2, 2, 2, 4, 5, 3, 2, 3, 3, 3, 3, 3, 4, 3, 3, 3, 5, 4, 6, 9, 9, 6, 4, 5, 3, 3, 3, 4, 4, 4, 3, 3, 3, 2, 3, 5, 5, 3, 3, 3, 4, 4, 3

Offset: 0

Rémy Sigrist, Mar 08 2020

A square-(sub)word consists of two nonempty identical adjacent subwords.
This sequence is a binary variant of A088950.
Square-subwords are counted with multiplicity.
A binary word of length 4 contains necessarily a square-subword, hence a(n) tends to infinity as n tends to infinity (a number whose binary representation has >= 4*k digits has >= k square-subwords).

			For n = 43:
- the binary representation of 43 is "101011",
- we have the following square-subwords: "1010", "0101", "11",
- hence a(43) = 3.

Rémy Sigrist, Table of n, a(n) for n = 0..16384
Eric Weisstein's World of Mathematics, Squarefree Word
Index entries for sequences related to binary expansion of n

Cf. A002620, A088950.

a(n, base=2) = { my (b=digits(n, base), v); for (w=1, #b\2, for (i=1, #b-2*w+1, if (b[i..i+w-1]==b[i+w..i+2*w-1], v++))); return (v) }

a(2^k) = a(2^k-1) = A002620(k) for any k >= 0.

A088951 Number of distinct square-subwords in ternary representation of n.

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, Oct 25 2003

nonn

a(n) <= A088950(n).

			n=125: a(125)=2 because 125 -> '11122' has 3 square-subwords: 11, 11 and 22 (11---, -11-- and ---22) and two of them are distinct.

Eric Weisstein's World of Mathematics, Squarefree Word

Cf. A007089.

cp's OEIS Frontend

A088952 Numbers that are squarefree words in ternary representation.

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A333124 a(n) is the number of square-subwords in the binary representation of n.

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PARI

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A088951 Number of distinct square-subwords in ternary representation of n.

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