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

0,2

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

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

Cf. A001317, A007089, A006943, A083093.

Sequence in context: A352205 A364582 A227434 * A366814 A031003 A324784

Adjacent sequences: A173016 A173017 A173018 * A173020 A173021 A173022

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