A088917 Central Delannoy numbers (mod 3); Characteristic function for Cantor set.

1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Author

Benoit Cloitre, Nov 30 2003

Keywords

• nonn

Comments

Also Apery numbers (mod 3).

More generally also (Sum_{k=0..n} binomial(n,k)^x*binomial(n+k,k)^y) (mod 3) for any x >= 1 in N and any odd y >= 1.

a(n) = 0 if the ternary expansion of n contains one or more 1-digits, otherwise 1. - Antti Karttunen, Aug 23 2019

Main diagonal of the Sierpinski carpet (A153490). - Paolo Xausa, May 19 2023

Links

• Antti Karttunen, Table of n, a(n) for n = 0..59048
• Michael Coons and James Evans, A sequential view of self-similar measures, or, What the ghosts of Mahler and Cantor can teach us about dimension, arXiv:2011.10722 [math.NT], 2020. See Figure 2 p. 2.
• Eric Weisstein's World of Mathematics, Cantor Fractal.
• Index entries for characteristic functions

Crossrefs

Cf. A005258, A081606, A001850, A105220, A153490, A316829.

Characteristic function of A005823, and with offset 1, characteristic function of A191106.

Programs

Mathematica

Nest[ Flatten[# /. {0 -> {0, 0, 0}, 1 -> {1, 0, 1}}] &, {1}, 5] (* Or *)
f[n_] := Mod[LegendreP[n, 3], 3]; Array[f, 111, 0] (* Or *)
f[n_] := If[ FreeQ[ IntegerDigits[n, 3], 1], 1, 0]; Array[f, 111, 0] (* also from Mathematica v8.0 Mathematical Functions Help section for "IntegerDigits" "Cantor set construction:" *) (* Robert G. Wilson v, Jun 16 2011 *)
Nest[Join[#, 0 #, #] &, {1}, 5] (* IWABUCHI Yu(u)ki, Sep 08 2012 *)

PARI

a(n)=sum(k=0,n,binomial(n,k)*binomial(n+k,k))%3

PARI

A088917(n) = { while(n, if(n%3==1, return(0), n\=3)); (1); }; \\ Antti Karttunen, Aug 23 2019 (copied from A005823)

PARI

A088917(n) = abs(factorback(apply(d -> d-1,digits(n,3)))); \\ Antti Karttunen, Aug 23 2019

Formula

a(A005823(n)) = 1; a(A081606(n)) = 0.

a(n) = A001850(n) - 3*floor(A001850(n)/3).

a(n) = 2 - A105220(n) = 1 - A316829(n). - Antti Karttunen and Jon Maiga, Aug 24 2019

G.f.: Product_{k>=0} (1 + x^(2*3^k)). - Ilya Gutkovskiy, Jun 05 2021

Extensions

Secondary name added by Antti Karttunen, Aug 23 2019