OFFSET
0,2
COMMENTS
Previous name was "Pascal's Triangle mod 3 converted to decimal."
If 2|a(n), then 4|a(n).
If 8|a(n), then 16|a(n).
If a(n)=4*a(n-1), then 3 does not divide n.
The first few odd values for a(n) are a(0)=1, a(8)=12301, a(20)=6199180549, a(24)=303766932781.
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
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
P. Mathonet, M. Rigo, M. Stipulanti and N. Zénaïdi, On digital sequences associated with Pascal's triangle, arXiv:2201.06636 [math.NT], 2022.
FORMULA
a(3^n) = 3^(3^n) + 1.
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).]
For n > 0, a((3^n)+1) = 4*a(3^n) and a((3^n)+2) = 4*a((3^n)+1).
a(n) = Sum_{k=0..n} A083093(n,k) * 3^k. - Reinhard Zumkeller, Jul 11 2013
EXAMPLE
a(9) = 3^(3^2) + 1 = 19684;
a(8) = (5*19684 - 12)/8 = 12301;
a(10) = 4*19684 = 78736.
MATHEMATICA
FromDigits[#, 3] & /@ Table[Mod[Binomial[n, k], 3], {n, 0, 25}, {k, 0, n}] (* Michael De Vlieger, Oct 31 2018 *)
PROG
(Haskell)
a173019 = foldr (\t v -> 3 * v + t) 0 . map toInteger . a083093_row
-- Reinhard Zumkeller, Jul 11 2013
(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
CROSSREFS
KEYWORD
base,easy,nonn
AUTHOR
Michael Thaler (michael_thaler(AT)brown.edu), Nov 07 2010
EXTENSIONS
a(13) and a(19) corrected and name clarified by Tom Edgar, Oct 11 2015
STATUS
approved