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

0,1

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

We can consider each occurrence of a(n) = 1 to represent an image of the Cantor middle thirds set spanning [n, n+1], with a(m) = 0 indicating that [m, m+1] is empty. Then for k >= 0, the finite sequence a(0) .. a(3^k-1) represents a scaled image of the middle thirds set spanning [0, 3^k], illustrating a symmetry of scale with respect to the endpoint of the set. For scale symmetry around the set's quarter points see A355681 and A355682. - Peter Munn, Sep 19 2025

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.

Steven Robertson and Noy Soffer Aranov, Fractals Emerging from the Toepltiz Determinants of the p-Cantor Sequence, arXiv:2510.19449 [math.NT], 2025. See p. 6.

Eric Weisstein's World of Mathematics, Cantor Fractal.

Index entries for characteristic functions

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

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 *)

Prog

(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

Crossrefs

Cf. A005258, A081606, A001850, A105220, A153490, A316829, A355681, A355682.

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

Sequence in context: A257585 A285034 A266174 * A014933 A011643 A015941

Adjacent sequences: A088914 A088915 A088916 * A088918 A088919 A088920

Keyword

nonn,easy

Author

Benoit Cloitre, Nov 30 2003

Extensions

Secondary name added by Antti Karttunen, Aug 23 2019

Status

approved