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

%I A088917 #60 Feb 16 2025 08:32:51
%S A088917 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,
%T A088917 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,
%U A088917 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
%N A088917 Central Delannoy numbers (mod 3); Characteristic function for Cantor set.
%C A088917 Also Apery numbers (mod 3).
%C A088917 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.
%C A088917 a(n) = 0 if the ternary expansion of n contains one or more 1-digits, otherwise 1. - _Antti Karttunen_, Aug 23 2019
%C A088917 Main diagonal of the Sierpinski carpet (A153490). - _Paolo Xausa_, May 19 2023
%H A088917 Antti Karttunen, <a href="/A088917/b088917.txt">Table of n, a(n) for n = 0..59048</a>
%H A088917 Michael Coons and James Evans, <a href="https://arxiv.org/abs/2011.10722">A sequential view of self-similar measures, or, What the ghosts of Mahler and Cantor can teach us about dimension</a>, arXiv:2011.10722 [math.NT], 2020. See Figure 2 p. 2.
%H A088917 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CantorSet.html">Cantor Fractal</a>.
%H A088917 <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>
%F A088917 a(A005823(n)) = 1; a(A081606(n)) = 0.
%F A088917 a(n) = A001850(n) - 3*floor(A001850(n)/3).
%F A088917 a(n) = 2 - A105220(n) = 1 - A316829(n). - _Antti Karttunen_ and _Jon Maiga_, Aug 24 2019
%F A088917 G.f.: Product_{k>=0} (1 + x^(2*3^k)). - _Ilya Gutkovskiy_, Jun 05 2021
%t A088917 Nest[ Flatten[# /. {0 -> {0, 0, 0}, 1 -> {1, 0, 1}}] &, {1}, 5] (* Or *)
%t A088917 f[n_] := Mod[LegendreP[n, 3], 3]; Array[f, 111, 0] (* Or *)
%t A088917 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 *)
%t A088917 Nest[Join[#, 0 #, #] &, {1}, 5] (* _IWABUCHI Yu(u)ki_, Sep 08 2012 *)
%o A088917 (PARI) a(n)=sum(k=0,n,binomial(n,k)*binomial(n+k,k))%3
%o A088917 (PARI) A088917(n) = { while(n, if(n%3==1, return(0), n\=3)); (1); }; \\ _Antti Karttunen_, Aug 23 2019 (copied from A005823)
%o A088917 (PARI) A088917(n) = abs(factorback(apply(d -> d-1,digits(n,3)))); \\ _Antti Karttunen_, Aug 23 2019
%Y A088917 Cf. A005258, A081606, A001850, A105220, A153490, A316829.
%Y A088917 Characteristic function of A005823, and with offset 1, characteristic function of A191106.
%K A088917 nonn
%O A088917 0,1
%A A088917 _Benoit Cloitre_, Nov 30 2003
%E A088917 Secondary name added by _Antti Karttunen_, Aug 23 2019