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

1, 4, 16, 28, 112, 448, 784, 3136, 12301, 19684, 78736, 314944, 551152, 2204608, 8818432, 15432256, 61729024, 242132884, 387459856, 1549839424, 6199180549, 10848875968, 43395503872, 173577055372, 303766932781, 1215067731124

Offset: 0

Author

Michael Thaler (michael_thaler(AT)brown.edu), Nov 07 2010

Keywords

• base
• easy
• nonn

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

Subsequence of A014190. - Chai Wah Wu, Jul 30 2025

Examples

			a(9) = 3^(3^2) + 1 = 19684;
a(8) = (5*19684 - 12)/8 = 12301;
a(10) = 4*19684 = 78736.
		

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.

Crossrefs

Cf. A006940 (takes these values and converts them to decimal notation).

Cf. A001317, A007089, A006943, A083093, A014190.

Programs

Haskell

a173019 = foldr (\t v -> 3 * v + t) 0 . map toInteger . a083093_row
-- Reinhard Zumkeller, Jul 11 2013

Mathematica

FromDigits[#, 3] & /@ Table[Mod[Binomial[n, k], 3], {n, 0, 25}, {k, 0, n}] (* Michael De Vlieger, Oct 31 2018 *)

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

Python

from math import prod, comb
from gmpy2 import digits
def A173019(n):
    if n==0: return 1
    c, l = '', len(s:=digits(n,3))
    for k in range(m:=n+2>>1):
        t = digits(k,3).zfill(l)
        c += str(prod(comb(int(s[i]),int(t[i]))%3 for i in range(l))%3)
    return int(c+c[m-2+(n&1)::-1],3) # Chai Wah Wu, Jul 30 2025

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

Extensions

a(13) and a(19) corrected and name clarified by Tom Edgar, Oct 11 2015