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A173019 a(n) is the value of row n in triangle A083093 seen as ternary number.

%I A173019 #59 Aug 01 2025 00:09:35
%S A173019 1,4,16,28,112,448,784,3136,12301,19684,78736,314944,551152,2204608,
%T A173019 8818432,15432256,61729024,242132884,387459856,1549839424,6199180549,
%U A173019 10848875968,43395503872,173577055372,303766932781,1215067731124
%N A173019 a(n) is the value of row n in triangle A083093 seen as ternary number.
%C A173019 Previous name was "Pascal's Triangle mod 3 converted to decimal."
%C A173019 If 2|a(n), then 4|a(n).
%C A173019 If 8|a(n), then 16|a(n).
%C A173019 If a(n)=4*a(n-1), then 3 does not divide n.
%C A173019 The first few odd values for a(n) are a(0)=1, a(8)=12301, a(20)=6199180549, a(24)=303766932781.
%C A173019 It appears that, as the terms of A001317 (analogous to this sequence, using binary instead of ternary) can be uniquely represented as products of Fermat numbers, the terms of this sequence can be represented as products from a nontrivial set of numbers. - _Thomas Anton_, Oct 27 2018
%C A173019 Subsequence of A014190. - _Chai Wah Wu_, Jul 30 2025
%H A173019 Reinhard Zumkeller, <a href="/A173019/b173019.txt">Table of n, a(n) for n = 0..1000</a>
%H A173019 P. Mathonet, M. Rigo, M. Stipulanti and N. Zénaïdi, <a href="https://arxiv.org/abs/2201.06636">On digital sequences associated with Pascal's triangle</a>, arXiv:2201.06636 [math.NT], 2022.
%F A173019 a(3^n) = 3^(3^n) + 1.
%F A173019 a(3^n) = (8*a((3^n)-1) + 12)/5. [5*a(3^n) = 1200...0012 (base 3), 8*a((3^n)-1) = (22)(1212...2121) = 11222...2202 (base 3).]
%F A173019 For n > 0, a((3^n)+1) = 4*a(3^n) and a((3^n)+2) = 4*a((3^n)+1).
%F A173019 a(n) = Sum_{k=0..n} A083093(n,k) * 3^k. - _Reinhard Zumkeller_, Jul 11 2013
%e A173019 a(9) = 3^(3^2) + 1 = 19684;
%e A173019 a(8) = (5*19684 - 12)/8 = 12301;
%e A173019 a(10) = 4*19684 = 78736.
%t A173019 FromDigits[#, 3] & /@ Table[Mod[Binomial[n, k], 3], {n, 0, 25}, {k, 0, n}] (* _Michael De Vlieger_, Oct 31 2018 *)
%o A173019 (Haskell)
%o A173019 a173019 = foldr (\t v -> 3 * v + t) 0 . map toInteger . a083093_row
%o A173019 -- _Reinhard Zumkeller_, Jul 11 2013
%o A173019 (PARI) a(n) = my(v = vector(n+1, k, binomial(n, k-1))); fromdigits(apply(x->x % 3, v), 3); \\ _Michel Marcus_, Nov 21 2018
%o A173019 (Python)
%o A173019 from math import prod, comb
%o A173019 from gmpy2 import digits
%o A173019 def A173019(n):
%o A173019     if n==0: return 1
%o A173019     c, l = '', len(s:=digits(n,3))
%o A173019     for k in range(m:=n+2>>1):
%o A173019         t = digits(k,3).zfill(l)
%o A173019         c += str(prod(comb(int(s[i]),int(t[i]))%3 for i in range(l))%3)
%o A173019     return int(c+c[m-2+(n&1)::-1],3) # _Chai Wah Wu_, Jul 30 2025
%Y A173019 Cf. A006940 (takes these values and converts them to decimal notation).
%Y A173019 Cf. A001317, A007089, A006943, A083093, A014190.
%K A173019 base,easy,nonn
%O A173019 0,2
%A A173019 Michael Thaler (michael_thaler(AT)brown.edu), Nov 07 2010
%E A173019 a(13) and a(19) corrected and name clarified by _Tom Edgar_, Oct 11 2015