A182929 The rows of the binomial triangle reduced to balanced ternary lists encoded as decimal numbers.

0, 1, 4, 7, 16, 61, 160, 547, 1456, 5110, 13120, 44287, 118096, 398581, 1075840, 3720094, 9565936, 32285041, 86093440, 290565367, 774840976, 2711943430, 7059662080, 23535794707, 61987278400, 212693848522, 564945153280, 1979718703900, 5083731656656

Offset

0,3

Comments

Define an operation ~: ZxZ -> {-1,0,1} (Z integers) by b ~ n = (b sigmod n) [|b sigmod n|=1]. Here [] is the Iverson bracket and sigmod is the signed mod operation defined as b sigmod n = b - n*ceil(b/n - 1/2) if n <> 0 and b otherwise. Further let T(n) = list_{k=0..n-1} binomial(n-1,k) ~ n for n > 1 and n if n is 0 or 1. We call T(n) the binomial notation of n.

A non-obvious arithmetical property of the binomial triangle becomes apparent from these balanced ternary lists: the rows which have an odd prime number as an index and only these rows are represented by a ternary list where 1 and -1 are alternating. One might also say that an odd integer is prime iff n > 1 and its binomial notation is zerofree.

Finally a(n) = Sum_{k=0..n-1} T(n)[k]*3^k.

The sequence starts at n=0 although the definitions can be extended to the negative integers by flipping the signs of the ternary digits. To illustrate the definitions:

n T(n) a(n)

----------------------------------

[-6] -1, 1, 0, 0, 1, -1 [-160]

[-5] -1, 1, -1, 1, -1 [ -61]

[-4] -1, 1, 1, -1 [ -16]

[-3] -1, 1, -1 [ -7]

[-2] -1, -1 [ -4]

[-1] -1 [ -1]

[ 0] 0 [ 0]

[ 1] 1 [ 1]

[ 2] 1, 1 [ 4]

[ 3] 1, -1, 1 [ 7]

[ 4] 1, -1, -1, 1 [ 16]

[ 5] 1, -1, 1, -1, 1 [ 61]

[ 6] 1, -1, 0, 0, -1, 1 [ 160]

Links

Table of n, a(n) for n=0..28.

Wikipedia, Balanced ternary

Maple

A182929 := proc(n) local lop, k, Tlist;
lop := proc(a, n) if n = 0 then a else mods(a, n);
`if`(abs(%)=1, %, 0) fi end;
Tlist := proc(n) `if`(abs(n)<2, n, seq(signum(n)*
lop(binomial(abs(n)-1, k), n), k=0..abs(n)-1)) end:
[Tlist(n)]; signum(n)*add(3^k*%[k+1], k=0..abs(n)-1) end:
seq(A182929(n), n=0..30);

Mathematica

lop[a_, n_] := Module[{m}, m = If[n == 0, a, Mod[a, n, -Quotient[Abs[n]-1, 2]]]; If[Abs[m] == 1, m, 0]];
Tlist[n_] := If[Abs[n]<2, {n}, Table[Sign[n]*lop[Binomial[Abs[n]-1, k], n], {k, 0, Abs[n]-1}]];
a[n_] := Module[{t=Tlist[n]}, Sign[n]*Sum[3^k*t[[k+1]], {k, 0, Abs[n]-1}]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 22 2019, from Maple *)

Prog

(PARI)
a(n) = {
  if (n == 0, 0,
  subst(apply(r->if(r==1, 1, r==n-1, -1, 0), lift(Mod(1+'x, n)^(n-1))), 'x, 3));
};
vector(29, n, a(n-1))  \\ Gheorghe Coserea, Nov 21 2016

Crossrefs

Cf. A007318, A047999, A001317.

Sequence in context: A246915 A340600 A013625 * A361733 A367744 A363642

Adjacent sequences: A182926 A182927 A182928 * A182930 A182931 A182932

Keyword

nonn

Author

Peter Luschny, Mar 16 2011

Status

approved