A033564 -id:A033564

Search: a033564 -id:a033564

Displaying 1-3 of 3 results found.

page 1

A018238

Add 1 to leading digit and put in front.

+10
3

1, 21, 3121, 41213121, 5121312141213121, 61213121412131215121312141213121, 7121312141213121512131214121312161213121412131215121312141213121 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,2`
	COMMENTS	`The concatenation of first n terms (if n is small) yields a palindrome: 1, 121, 1213121, etc. - Amarnath Murthy, Apr 08 2003` `From M. F. Hasler, May 05 2008: (Start)` `This is not the case from n=10 on: According to the formula in A123121 A082215(10) has an even number of digits, the middle digits being "10". (In a strict sense, e.g. Def. 3 of the first reference there, A082215(9) is the last Zimin word on the alphabet {1,...,9}, though.)` `While there is less ambiguity about the definition of A018238(10), it is not clear if A018238(11) should start with "11..." or with "10..." (the largest digit of all subsequent terms being "9"). According to the formula in A123121, a(100) has 3 digits more than a(99), so the first choice seems appropriate and has been adopted for the given PARI code.` `However, it corresponds to a modified definition, "a(n) = concatenation of n and all preceding terms". a(3) is the only prime term up to a(14) included. The sequence is (1,0,1,0,1,0,...) (mod 3), at least up to a(20). (End)`
	LINKS	`M. F. Hasler, Table of n, a(n) for n=1,...,11.`
	PROG	`(PARI) A018238(n, t="")=for(k=2, n, t=Str(t, k-1, t)); eval(Str(n, t)) \\ M. F. Hasler, May 05 2008`
	CROSSREFS	`Cf. A001511, A082215, A123121, A033564.`
	KEYWORD	`nonn,base`
	AUTHOR	`N. J. A. Sloane, Michael Minic (minic(AT)mtsu.edu)`
	STATUS	`approved`

A082215

Concatenation of terms of A018238.

+10
2

1, 121, 1213121, 121312141213121, 1213121412131215121312141213121, 121312141213121512131214121312161213121412131215121312141213121, 1213121412131215121312141213121612131214121312151213121412131217121312141213121512131214121312161213121412131215121312141213121 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,2`
	COMMENTS	`Also called Zimin words.` `a(n) is a palindrome for n<10; it is debatable whether a(n) can be called a Zimin word for n>=10 (see the Comments in A018238). - Danny Rorabaugh, Sep 26 2015`
	LINKS	`Table of n, a(n) for n=1..7.` `J. Cooper and D. Rorabaugh, Bounds on Zimin Word Avoidance, arXiv:1409.3080 [math.CO], 2014; Congressus Numerantium, 222 (2014), 87-95.` `L. J. Cummings and M. Mays, A one-sided Zimin construction, Electron. J. Combin. 8 (2001), #R27.` `A. I. Zimin, Blocking sets of terms, Math. USSR Sbornik, 47 (1984), No. 2, 353-364.`
	FORMULA	`The Zimin words are defined here by Z_1 = 1, Z_n = Z_{n-1}nZ_{n-1}. - Dmitry Kamenetsky, Sep 30 2006`
	MATHEMATICA	`a = {1}; Do[w = IntegerDigits@ a[[n - 1]]; AppendTo[a, FromDigits@ Join[w, IntegerDigits@ n, w]], {n, 2, 7}]; a (* Michael De Vlieger, Sep 26 2015 *)`
	CROSSREFS	`Cf. A018238, A123121, A033564.` `See A001511 for another representation of this sequence of digits.`
	KEYWORD	`base,nonn`
	AUTHOR	`Amarnath Murthy, Apr 08 2003`
	EXTENSIONS	`More terms from Joshua Zucker, May 08 2006` `"Palindromes" replaced with "Numbers" in sequence name by Danny Rorabaugh, Sep 26 2015` `Shorter name by Joerg Arndt, Aug 28 2021`
	STATUS	`approved`

A348466

a(0)=1; for n >= 0, a(n+1) is the concatenation of a(n) and a(n)-1.

+10
0

1, 10, 109, 109108, 109108109107, 109108109107109108109106, 109108109107109108109106109108109107109108109105, 109108109107109108109106109108109107109108109105109108109107109108109106109108109107109108109104 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,2`
	LINKS	`Table of n, a(n) for n=0..7.`
	MATHEMATICA	`Nest[Append[#1, #210^IntegerLength[#3] + #3] & @@ {#, #[[-1]], #[[-1]] - 1} &, {1, 10}, 6] ( Michael De Vlieger, Oct 20 2021 *)`
	CROSSREFS	`Suggested by A033564.`
	KEYWORD	`nonn,base`
	AUTHOR	`N. J. A. Sloane, Oct 20 2021`
	STATUS	`approved`

page 1

Search completed in 0.006 seconds